{"id":336,"date":"2017-08-28T04:31:11","date_gmt":"2017-08-28T04:31:11","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=336"},"modified":"2017-09-25T16:50:35","modified_gmt":"2017-09-25T16:50:35","slug":"differential-equations-sequences-and-series","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/differential-equations-sequences-and-series\/","title":{"rendered":"Differential Equations, Sequences, and Series"},"content":{"rendered":"
\n
\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a><\/div>\n

<\/p>\n

Differential Equations, Sequences, and Series<\/h1>\n

Objective<\/h4>\n

In this lesson, you will learn how to solve and apply elementary differential equations, convergence and divergence\u00a0of sequences and series, differentiation and integration of power series, and Taylor series expansions of basic\u00a0functions.<\/p>\n

Previously Covered:<\/h4>\n
    \n
  • A function\u00a0\u00a0is the antiderivative <\/em><\/strong>or indefinite\u00a0integral<\/em><\/strong> of\u00a0\u00a0if\u00a0\u00a0for all\u00a0\u00a0in an interval where\u00a0\u00a0is defined and\u00a0\u00a0is continuous and differentiable. The antiderivative\u00a0\u00a0is a function rather than a number.<\/li>\n
  • If a function is continuous and\u00a0integrable on\u00a0,\u00a0then the definite integral\u00a0\u00a0<\/strong>is a number rather than a function.<\/li>\n
  • The Fundamental Theorem of Calculus <\/em><\/strong>relates\u00a0the derivative of the indefinite integral and the integrand\u00a0function, and shows that the value of the definite integral is\u00a0the difference in the value of the antiderivative at the\u00a0endpoints of the interval of integration.<\/li>\n<\/ul>\n
    \n

    What is a differential equation?<\/h3>\n

    A differential\u00a0equation<\/abbr> is an equation that contains derivatives. The order\u00a0of a differential equation is the order of the highest ordered\u00a0derivative in the equation. Thus,\u00a0\u00a0is a first-order differential equation, while\u00a0\u00a0is a second-order differential equation. A thorough study of\u00a0differential equations is typically reserved for advanced\u00a0calculus. This lesson examines elementary differential equations\u00a0of the forms\u00a0and\u00a0,\u00a0where\u00a0\u00a0is a constant.<\/p>\n

    Suppose\u00a0\u00a0represents a quantity such as a population of bacteria, the volume\u00a0of a dye used in a medical analysis, the mass of a decaying\u00a0radioactive substance, the future value of a compounded\u00a0investment, etc. Then, the quantity\u00a0\u00a0represents the growth of\u00a0.\u00a0In these and other applications, the relative rate of growth is\u00a0proportional to its quantity; that is,\u00a0,\u00a0where\u00a0\u00a0is the constant rate of increase\u00a0\u00a0or decrease\u00a0\u00a0in the quantity. Rewriting the equation as\u00a0,\u00a0we find\u00a0,\u00a0or\u00a0.\u00a0Integrating both sides, we get\u00a0.\u00a0Therefore,\u00a0.\u00a0This equation has the solution\u00a0\u00a0where\u00a0\u00a0is a positive constant. Therefore,\u00a0\u00a0where\u00a0\u00a0can be any real number. This equation describes exponential\u00a0growth<\/abbr> and decay<\/abbr>,\u00a0which is represented graphically below for\u00a0\u00a0and\u00a0\u00a0respectively.<\/p>\n\n\n\n\n
    \"Exponential<\/td>\n\"Exponential<\/td>\n<\/tr>\n
    <\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    There are an infinite\u00a0number of solutions to this differential equation, as the choices\u00a0for\u00a0\u00a0and\u00a0\u00a0are infinite. Typical physical problems, however, have a single\u00a0solution. This is rectified by specifying <\/span>initial\u00a0conditions<\/span><\/em>, such\u00a0as\u00a0\u00a0where\u00a0\u00a0and\u00a0\u00a0are given or measured. <\/span><\/p>\n

    Differential Equations, Sequences, and Series<\/h3>\n

    Let’s try some examples. The rate of growth of a bacteria\u00a0culture is proportional to its population. Initially, the\u00a0population is 10,000, but it increases to 25,000 after 10 days.\u00a0What is the size of the population after 20 days?<\/p>\n

    This is an exponential growth problem of the form\u00a0,\u00a0where\u00a0\u00a0is the population after t<\/em>\u00a0 days, and\u00a0.\u00a0The general solution is\u00a0.\u00a0At this point, we have an equation for the population with two\u00a0unknowns. To find a unique solution, we are given two initial\u00a0conditions: (i)\u00a0,\u00a0and (ii)\u00a0.\u00a0Initial condition (i) gives us\u00a0\u00a0or\u00a0\u00a0Initial condition (ii) gives us:\u00a0\u00a0Therefore,\u00a0\u00a0Taking the natural log of both sides.\u00a0,\u00a0so\u00a0.\u00a0Therefore,<\/p>\n

    \u00a0is the unique solution for the population at any time t<\/em>.\u00a0After 20 days,<\/p>\n

    <\/p>\n

    Newton\u2019s law of cooling says that the rate of change in\u00a0temperature\u00a0\u00a0of an object is proportional to \u00a0the temperature difference between\u00a0the object and its surroundings. If the constant of\u00a0proportionality is\u00a0,\u00a0and the initial temperature is\u00a0\u00a0(A) find an expression for\u00a0\u00a0in terms of\u00a0\u00a0and\u00a0,\u00a0and (B) in terms of\u00a0,\u00a0how much time does it take for the temperature difference to be\u00a0cut in half?<\/p>\n

    (A) Over time, the temperature of the object approaches the\u00a0temperature of the surroundings; therefore, the difference\u00a0decreases over time, so this is an example of exponential decay.\u00a0We note\u00a0\u00a0for\u00a0,\u00a0so\u00a0.\u00a0Our initial\u00a0\u00a0condition is\u00a0\u00a0so our expression for\u00a0\u00a0is<\/p>\n

    .<\/p>\n

    (B) We want to find the time\u00a0\u00a0for which\u00a0.\u00a0We find\u00a0,\u00a0so\u00a0.\u00a0Therefore,<\/p>\n

    <\/p>\n

    Differential Equations, Sequences, and Series<\/h3>\n

    The next differential equation we examine is the form\u00a0or\u00a0,\u00a0where\u00a0\u00a0is a constant. As a guess, assume a solution\u00a0.\u00a0We find:<\/p>\n

    ,<\/p>\n

    or<\/p>\n

    ,<\/p>\n

    which satisfies the differential equation when\u00a0.\u00a0Although we chose a cosine function, this also applies to a sine\u00a0function. The\u00a0\u00a0variable allows for the shift from cosine to sine. Since the\u00a0cosine function repeats itself over time, this differential\u00a0equation describes periodic motion<\/em> or oscillations. We are\u00a0typically interested in periodic and bounded motion, known as\u00a0harmonic\u00a0motion<\/abbr>.<\/p>\n

    The\u00a0-term\u00a0is known as the angular frequency<\/em>, measured in radian\/sec.\u00a0The term\u00a0\u00a0is called the phase of motion<\/em>, and\u00a0\u00a0is the phase constant. If we imagine an object moving in a circle\u00a0of radius\u00a0\u00a0about the origin,\u00a0\u00a0is the angular displacement\u00a0from the <\/span>x-<\/span><\/em>axis\u00a0at\u00a0,\u00a0and\u00a0\u00a0is the angular speed of the object. <\/span><\/p>\n

    Here is an example.\u00a0Newton\u2019s second law of classical mechanics states\u00a0that a force\u00a0\u00a0acting on a constant mass\u00a0\u00a0is proportional to the acceleration\u00a0\u00a0of the mass,\u00a0.\u00a0Previously, we learned that acceleration is the second derivative\u00a0of the position function; that is,\u00a0\u00a0(assuming we restrict motion\u00a0to the <\/span>x-<\/span><\/em>direction).\u00a0We also learned of Hooke\u2019s law, which states that the force\u00a0exerted by a spring that is displaced a distance\u00a0\u00a0from its natural position is\u00a0,\u00a0where\u00a0\u00a0is a constant. Suppose a mass \u00a0<\/span>\u00a0attached to the end of a massless spring of spring-constant\u00a0\u00a0is placed on a frictionless surface and is initially stretched a\u00a0distance\u00a0\u00a0from its natural position. (A) Show that the behavior of the\u00a0spring-mass system is described by harmonic motion, and (B) find\u00a0the angular frequency in terms of\u00a0\u00a0and\u00a0.<\/p>\n

    (A) Setting both force equations equal to one another, we\u00a0find\u00a0,\u00a0or\u00a0\u00a0Since\u00a0\u00a0and\u00a0\u00a0are both positive constants, the ratio\u00a0\u00a0is positive. This is in the form\u00a0,\u00a0and is therefore an equation for harmonic motion.<\/p>\n

    (B) From the equation\u00a0,\u00a0we know\u00a0<\/p>\n

    What are sequences and series?<\/h3>\n

    A sequence\u00a0<\/abbr>is a function whose domain is the set of positive\u00a0integers. Recall that the domain<\/em> of a function\u00a0\u00a0is the set of all possible values of\u00a0.\u00a0For example, the sequence\u00a0\u00a0has the reciprocals of all positive integers as its elements<\/em>.<\/p>\n

    If\u00a0\u00a0is a sequence, then\u00a0\u00a0is known as a series<\/abbr>.\u00a0Each\u00a0\u00a0is a term<\/em> of the series. Of special interest is an infinite\u00a0series<\/em>, where the number of terms is infinite. A geometric\u00a0series<\/em> is an infinite series where consecutive terms differ by\u00a0the same ratio; the simplest form is\u00a0,\u00a0where\u00a0\u00a0is constant. A power series <\/em>is of the form\u00a0\u00a0(specifically, a \u201cpower series in\u00a0\u201d),\u00a0and it will be given special treatment in this module.<\/p>\n

    Sequences and series can be convergent<\/em> or\u00a0divergent<\/em>. The sequence\u00a0\u00a0is <\/span>convergent<\/span><\/abbr> if \u00a0for some finite value\u00a0.\u00a0If\u00a0\u00a0or cannot be defined (for example, if the sequence eventually\u00a0settles into a pattern of oscillation between two distinct values)\u00a0, the sequence is <\/span>divergent<\/span><\/abbr>.\u00a0Likewise, an infinite series is convergent if the sum approaches a\u00a0finite value, and is divergent if the sum blows up to\u00a0
    \nor cannot be defined. It is easily proven that, if an infinite\u00a0series\u00a0<\/span>\u00a0is convergent, then\u00a0\u00a0Are the following sequences convergent or divergent?<\/p>\n

    <\/p>\n

    (A)\u00a0\u00a0so we use L\u2019H\u00f4pital\u2019s rule.<\/p>\n

    <\/p>\n

    The sequence converges to\u00a0<\/p>\n

    (B)\u00a0\u00a0so the sequence is divergent.<\/p>\n

    (C) The sequence forever alternates between\u00a0\u00a0and\u00a0,\u00a0so it is divergent.<\/p>\n

    For what values of r<\/em> is the infinite geometric series\u00a0\u00a0convergent?<\/p>\n

    The series is convergent if and only if\u00a0\u00a0For\u00a0,\u00a0the limit blows up to\u00a0.\u00a0For\u00a0\u00a0the limit alternates between\u00a0\u00a0For\u00a0,\u00a0the limit approaches\u00a0.\u00a0For\u00a0,\u00a0the limit alternates between\u00a0.\u00a0Since none of these limits approach 0, the series is divergent for\u00a0\u00a0However, for the values\u00a0,\u00a0we can express\u00a0\u00a0as a fraction\u00a0,\u00a0where\u00a0\u00a0and\u00a0.\u00a0Therefore,\u00a0\u00a0since\u00a0\u00a0Therefore, the series is convergent for\u00a0.<\/p>\n

    Differential Equations, Sequences, and Series<\/h3>\n

    If an infinite series converges, there may or may not be\u00a0techniques for calculating the exact sum. For the case of the\u00a0example, we will later show that the sum converges to\u00a0\u00a0for\u00a0.\u00a0Furthermore, if\u00a0\u00a0and\u00a0\u00a0converge, then\u00a0\u00a0converges, and\u00a0\u00a0converges as well.<\/p>\n

    There are several tests to determine whether an infinite series\u00a0of the form\u00a0\u00a0is convergent or divergent. For the case of an infinite series consisting only of non-negative terms, we have three tests at our disposal: the <\/span>comparison test<\/span><\/em>, the <\/span>limit comparison test<\/span><\/em>,\u00a0and the <\/span>integral test<\/span><\/em>. <\/span><\/p>\n

    \n

    Comparison Test<\/h4>\n

    Let be a\u00a0series with for all k<\/em><\/p>\n

    (i) If is a\u00a0convergent series with and for all k<\/em>, then is\u00a0also convergent.<\/p>\n

    (ii) If is a\u00a0divergent series with and for all k<\/em>, then is\u00a0also divergent.<\/p>\n

    Limit comparison test:<\/strong><\/p>\n

    Let and be series with and \u00a0for all k<\/em><\/p>\n

    (i) If , then\u00a0either both converge or both diverge.<\/p>\n

    (ii) If and\u00a0converges, then converges.<\/p>\n

    (iii) If and\u00a0 diverges, then diverges.<\/p>\n

    Integral test:<\/strong><\/p>\n

    Let be a\u00a0function that is continuous, positive, and decreasing for all .\u00a0Then the infinite series <\/p>\n

    (i) converges if the improper integral converges, and<\/p>\n

    (ii) diverges if .<\/p>\n<\/div>\n

    An example of each convergence test should help illustrate\u00a0their power.<\/p>\n

    For the series\u00a0,\u00a0,\u00a0use the integral test to determine the values of\u00a0\u00a0for which the series converges and diverges.<\/p>\n

    First, we note that\u00a0\u00a0is continuous, positive, and decreasing for\u00a0.\u00a0For\u00a0,\u00a0we find<\/p>\n

    <\/p>\n

    Therefore, the series diverges for\u00a0.\u00a0For\u00a0,<\/p>\n

    <\/p>\n

    Therefore, the series converges for\u00a0.\u00a0Finally, for\u00a0,<\/p>\n

    <\/p>\n

    Therefore, the series diverges for\u00a0.\u00a0If we put it all together, we find that\u00a0\u00a0diverges for\u00a0\u00a0and converges for\u00a0.\u00a0The series for\u00a0\u00a0is known as the harmonic\u00a0series<\/span><\/em>.<\/span><\/p>\n

    Given the above conclusion that the harmonic series diverges,\u00a0use the comparison test to determine whether\u00a0\u00a0converges or diverges.<\/p>\n

    For\u00a0,\u00a0,\u00a0so\u00a0.\u00a0Since we know the harmonic series diverges, the comparison test\u00a0tells us that\u00a0\u00a0diverges as well.<\/p>\n

    Solve the example using the limit comparison test, again using\u00a0the harmonic series for comparison. Let\u00a0\u00a0and\u00a0.\u00a0We find,<\/p>\n

    <\/p>\n

    Therefore, part (iii) of the limit comparison test tells us\u00a0that\u00a0\u00a0diverges.<\/p>\n

    Differential Equations, Sequences, and Series<\/h3>\n

    The study of infinite series is not restricted to positive\u00a0terms. One common infinite series with both positive and negative\u00a0terms is known as an alternating\u00a0series<\/span><\/em>. I<\/span>t\u00a0has the form\u00a0. The following theorem is a test of convergence specifically for\u00a0the alternating series.<\/p>\n

    \n

    Important Tidbit<\/h4>\n

    If the numbers alternate positive and\u00a0negative, and\u00a0if for all positive integers ,\u00a0and , then the alternating series is\u00a0convergent.<\/p>\n<\/div>\n

    Determine if the alternating series\u00a0\u00a0is convergent or divergent.<\/p>\n

    Note that this series has nearly the same terms as the harmonic\u00a0series (which is divergent), but the signs alternate.\u00a0\u00a0and\u00a0,\u00a0and\u00a0\u00a0for\u00a0,\u00a0so the first condition applies. We find\u00a0,\u00a0so this alternating series is convergent.\u00a0The above series\u00a0\u00a0is convergent, but the harmonic series\u00a0\u00a0is divergent. An infinite series like this is said to be\u00a0<\/span>conditionally\u00a0convergent<\/span><\/em> if\u00a0\u00a0is convergent but\u00a0\u00a0is divergent. If both series are convergent, then\u00a0\u00a0is said to be <\/span>absolutely\u00a0convergent<\/span><\/em>. If a\u00a0series is absolutely convergent, it is also convergent. With this\u00a0understanding of absolute convergence, we introduce two new tests\u00a0for absolute convergence: the <\/span>ratio\u00a0test<\/span><\/em> and the <\/span>root\u00a0test<\/span><\/em>. <\/span><\/p>\n

    \n

    Ratio Test<\/h4>\n

    Let be an infinite series with for all\u00a0k<\/em>. Then<\/p>\n

    (i) if , the series is\u00a0absolutely convergent (hence, convergent).<\/p>\n

    (ii) if (including ), the\u00a0series is divergent.<\/p>\n

    (iii) if , the test is\u00a0inconclusive.<\/p>\n

    Root test:<\/strong><\/p>\n

    Let be an infinite series with for all\u00a0k<\/em>. Then<\/p>\n

    (i) if , the series is\u00a0absolutely convergent (hence convergent).<\/p>\n

    (ii) if , the series is\u00a0divergent.<\/p>\n

    (iii) if , the test is\u00a0inconclusive.<\/p>\n<\/div>\n

    Here are a few more examples.<\/h3>\n

    Using the ratio test, determine whether\u00a0\u00a0converges or diverges.<\/p>\n

    We find\u00a0.<\/p>\n

    Therefore, the series is convergent.<\/p>\n

    Using the root test, determine whether\u00a0\u00a0converges or diverges.<\/p>\n

    .\u00a0Applying L\u2019H\u00f4pital\u2019s rule,
    \n.\u00a0Therefore, the root test is inconclusive.<\/p>\n

    We previously defined a power series in\u00a0\u00a0as a series of the form<\/p>\n

    \u00a0. A power series in\u00a0\u00a0is of the form<\/p>\n

    \u00a0where\u00a0\u00a0is a real number. However, we can easily convert this to the first\u00a0power series by setting\u00a0\u00a0and defining a power series in\u00a0\u00a0Therefore, we restrict our analysis to the first form of power\u00a0series for the following section.<\/p>\n

    For the power series\u00a0,\u00a0we define the radius\u00a0of convergence<\/abbr> as a number\u00a0\u00a0such that s<\/em> converges if\u00a0\u00a0and diverges if\u00a0.\u00a0At\u00a0\u00a0the series may either converge or diverge. If\u00a0\u00a0converges only when\u00a0,\u00a0then\u00a0If\u00a0\u00a0converges for all real values of\u00a0\u00a0then\u00a0<\/p>\n

    The set of values of\u00a0\u00a0for which\u00a0\u00a0is convergent is known as the interval\u00a0of convergence<\/abbr>. If\u00a0\u00a0is known, the interval of convergence can be determined. If\u00a0,\u00a0the interval of convergence is 0. If\u00a0\u00a0the interval of convergence is\u00a0.\u00a0Finally, if\u00a0\u00a0it is necessary to check\u00a0.\u00a0Depending on the convergence or divergence at these two points,\u00a0the interval of convergence is\u00a0<\/p>\n

    We can determine the radius of convergence by applying either\u00a0the ratio test or root test to the\u00a0\u00a0portion of the series.<\/p>\n

    For the power series\u00a0,\u00a0what are the radius of convergence and interval of convergence?<\/p>\n

    Apply the ratio test.\u00a0<\/p>\n

    The ratio test tells us that if\u00a0,\u00a0then the series is convergent, and if\u00a0,\u00a0then the test is inconclusive. So\u00a0we know that if |<\/span>x<\/span><\/em>|\u00a0< 1, then the series is convergent, and if |<\/span>x<\/span><\/em>|\u00a0= 1, the series may or may not be convergent. We will need to\u00a0check <\/span>x <\/span><\/em>=\u00a01 and <\/span>x<\/span><\/em>=\u00a0\u20131 individually. Either way,\u00a0.
    \n<\/span><\/p>\n

    Now we check <\/span>\u00a0and\u00a0.\u00a0At\u00a0,\u00a0the series is\u00a0,\u00a0which converges by the alternating series theorem. At\u00a0,\u00a0the series is\u00a0,\u00a0which converges, as we saw in the example. Therefore, the interval\u00a0of convergence is\u00a0.<\/p>\n

    If a power series has an interval of convergence, we can define\u00a0a function\u00a0\u00a0to describe the power series for each\u00a0\u00a0in that interval,\u00a0.\u00a0In some cases,\u00a0\u00a0is a familiar function and is easy to derive. For example, we can\u00a0derive\u00a0\u00a0for the simple geometric series\u00a0. In the example, we showed that\u00a0\u00a0exists in the interval of convergence of\u00a0.\u00a0We compute\u00a0. Since s<\/em> is not infinite in the interval of convergence,\u00a0we can subtract the second term from the first. Note that all\u00a0terms except 1 are eliminated.<\/p>\n

    So,\u00a0.\u00a0Therefore,\u00a0,\u00a0or\u00a0\u00a0for\u00a0.\u00a0With this, we can define other functions. For example,\u00a0,\u00a0and\u00a0,\u00a0where\u00a0\u00a0still holds.<\/p>\n

    Differential Equations, Sequences, and Series<\/h3>\n

    Since a power series can be expressed as a function\u00a0\u00a0within its interval of convergence,\u00a0\u00a0exist within the same interval. For the differentiation\u00a0and integration of power series<\/strong>, we perform the\u00a0appropriate calculation\u2014term by term. If the power series\u00a0has a radius of convergence\u00a0,\u00a0then for\u00a0:<\/p>\n

    (i) is continuous.<\/p>\n

    (ii) exists.<\/p>\n

    (iii) exists.<\/p>\n

    Thus differentiating or integrating a power series forms a new series with the same radius of convergence. From this, we can\u00a0derive other interesting functions expressed as infinite series.<\/p>\n

    From\u00a0,\u00a0for\u00a0,\u00a0find the power series expression for\u00a0\u00a0and\u00a0\u00a0for\u00a0.<\/p>\n

    By differentiating, we find\u00a0<\/p>\n

    By integrating, we find\u00a0.\u00a0By substituting\u00a0,\u00a0we find\u00a0,\u00a0or\u00a0.\u00a0Therefore,\u00a0.<\/p>\n

    Prove that\u00a0<\/p>\n

    First, we need to find the interval of convergence. By applying\u00a0the ratio test, we find that\u00a0.\u00a0Thus the radius of convergence is\u00a0,\u00a0and the interval of convergence is\u00a0. Next, take the derivative of both sides of\u00a0<\/p>\n

    ,\u00a0where we substituted\u00a0.\u00a0Note that we find\u00a0.\u00a0The solution for this differential equation is\u00a0.\u00a0At\u00a0,\u00a0,\u00a0so\u00a0.\u00a0This completes the proof.<\/p>\n

    Since the function\u00a0is continuous and differentiable within the same interval of\u00a0convergence, it follows that higher order derivatives exist on the\u00a0same interval. Since the series is infinite, the function\u00a0\u00a0is infinitely differentiable<\/em> on the interval. Taking the\u00a0first few derivatives of the power series, we find<\/p>\n

    ,<\/p>\n

    ,<\/p>\n

    .<\/p>\n

    At this point, note the following patterns for\u00a0.<\/p>\n

    <\/p>\n

    Differential Equations, Sequences, and Series<\/h3>\n

    In general, we can define the coefficients\u00a0\u00a0as\u00a0,\u00a0and the power series in x<\/em> can be rewritten as<\/p>\n

    .<\/p>\n

    The above series is known as the\u00a0<\/span>Maclaurin series<\/span><\/em>.\u00a0Generally, we can define the power series in\u00a0,
    \n<\/span><\/p>\n

    <\/p>\n

    and take the same steps to show<\/p>\n

    <\/p>\n

    This series is known as the Taylor series<\/strong> of\u00a0\u00a0at\u00a0.\u00a0While the Maclaurin series has an interval of convergence\u00a0,\u00a0the Taylor series has an interval of convergence\u00a0.\u00a0A series that is taken to the\u00a0\u00a0order rather than to\u00a0\u00a0is called an\u00a0\u00a0degree Taylor polynomial<\/strong>. In calculations, we\u00a0typically use Taylor polynomials to attain a desired degree of\u00a0accuracy.<\/p>\n

    Suppose we define an\u00a0\u00a0degree Taylor polynomial as\u00a0.\u00a0We call the difference between the function\u00a0\u00a0and the Taylor polynomial the remainder term\u00a0<\/strong>\u00a0in the interval of convergence. Therefore,\u00a0.\u00a0It can be shown that\u00a0,\u00a0where each\u00a0\u00a0is between\u00a0\u00a0and\u00a0.<\/p>\n

    Find the Maclaurin series expansion for\u00a0.<\/p>\n

    Set\u00a0.\u00a0All higher order derivatives\u00a0,\u00a0so\u00a0.\u00a0Therefore,<\/p>\n

    <\/p>\n

    Note that this was proven in the example to be true for all\u00a0.\u00a0Strictly speaking, this is not a formal proof that\u00a0\u00a0has a series expansion. It simply shows that if\u00a0\u00a0has a Maclaurin series expansion, then this series must be it.<\/p>\n

    Find the Taylor series expansion for\u00a0\u00a0at\u00a0.<\/p>\n

    <\/p>\n

    The Taylor expansion is therefore<\/p>\n

    <\/p>\n

    Approximate the value of\u00a0\u00a0using the first 6 terms of the Taylor expansion. Compute the value\u00a0using a ten-digit calculator and determine why both answers are\u00a0exactly the same.<\/p>\n

    In radians,\u00a0. Using the above formula for\u00a0\u00a0at\u00a0,<\/p>\n

    <\/p>\n

    Using a ten-digit calculator, you should find the same value.\u00a0The reason for this high degree of accuracy can be found in the\u00a0remainder term. The error introduced by using the first six terms\u00a0is\u00a0,\u00a0which is<\/p>\n

    <\/p>\n

    This shows that the above result should be accurate to\u00a0\u00a0decimal places, which is better than a ten-digit calculator.<\/p>\n

    Further Reading in Calculus<\/h3>\n

    Calculus<\/em> (Ron Larson and Robert P. Hostetler): Houghton\u00a0Mifflin, 2003.<\/p>\n

    Calculus Demystified: A Self-teaching Guide<\/em> (Steven G.\u00a0Kranz): McGraw-Hill, 2003.<\/p>\n

    Calculus Made Easy: Being a Very-Simplest Introduction to\u00a0Those Beautiful Methods of Reckoning Which Are Generally Called by\u00a0the Terrifying Names of the Differential Calculus<\/em> (Silvanus\u00a0Philips Thompson and Martin Gardner): St. Martin’s Press, 1998.<\/p>\n

    How to Ace Calculus: The Streetwise Guide<\/em> and How to\u00a0Ace the Rest of Calculus: The Streetwise Guide<\/em>. (Colin Adams,\u00a0et. al.): Henry Holt and Company, 1998.<\/p>\n

    The Complete Idiot’s Guide to Calculus<\/em> (W. Michael\u00a0Kelley): Alpha Books, 2002.<\/p>\n

    Don’t forget to test your knowledge\u00a0with the Calculus<\/em> Chapter Quiz;<\/a><\/em><\/strong><\/p>\n<\/section>\n

    \u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a><\/div>\n

    Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

    \u2b05 Previous Lesson\u00a0Workshop Index Differential Equations, Sequences, and Series Objective In this lesson, you will learn how to solve and apply elementary differential equations, convergence and divergence\u00a0of sequences and series, differentiation and integration of power series, and Taylor series expansions of basic\u00a0functions. Previously Covered: A function\u00a0\u00a0is the antiderivative or indefinite\u00a0integral of\u00a0\u00a0if\u00a0\u00a0for all\u00a0\u00a0in an interval where\u00a0\u00a0is […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-336","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/336","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=336"}],"version-history":[{"count":12,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/336\/revisions"}],"predecessor-version":[{"id":832,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/336\/revisions\/832"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}