Unit 10: Infinite Sequences and Series

Example: since $1/[k(k+1)]=1/k-1/(k+1)$,
$$S_n=\sum_{k=1}^{n}\frac{1}{k(k+1)}=\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\cdots+\left(\frac1n-\frac1{n+1}\right)=1-\frac1{n+1}.$$
Therefore $\lim_{n\to\infty}S_n=1$, so $\sum_{k=1}^{\infty}1/[k(k+1)]$ converges and its sum is $1$. By contrast, for $\sum_{k=1}^{\infty}1$, $S_n=n\to\infty$, so the series diverges.

The finite partial sum through exponent $N$ is $S_N=a(1-r^{N+1})/(1-r)$ for $r\ne1$. When $|r|<1$, $r^{N+1}\to0$, leaving $a/(1-r)$. If $|r|\ge1$, that power does not approach zero, so the partial sums do not approach a finite value.

Example: $6-3+\tfrac32-\tfrac34+\cdots$ is geometric because each term is the previous term multiplied by $-1/2$. Here $a=6$ and $r=-1/2$. Since $|-1/2|<1$, the series converges, and
$$S=\frac{6}{1-(-1/2)}=\frac{6}{3/2}=4.$$
The alternating signs do not prevent convergence because the term magnitudes shrink.

Example 1: for $\sum_{n=1}^{\infty}n/(n+1)$,
$$\lim_{n\to\infty}\frac{n}{n+1}=1\ne0.$$
The terms keep contributing about $1$, so the nth term test proves that the series diverges. No other convergence test is needed.

Example 2: for $\sum_{n=1}^{\infty}1/n$, the term limit is $0$. The nth term test is inconclusive: it does not say the series converges. A different test is required to determine its behavior.

Example: for $\sum_{n=1}^{\infty}1/(n^2+1)$, let $f(x)=1/(x^2+1)$. For $x\ge1$, $f$ is continuous and positive, and $f'(x)=-2x/(x^2+1)^2<0$, so it is decreasing. Also,
$$\int_1^{\infty}\frac{dx}{x^2+1}=\lim_{b\to\infty}[\arctan x]_1^b=\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}.$$
The improper integral converges, so the series converges by the integral test.

• LIM-7.A Determine whether a series converges or diverges.
• LIM-7.A.7 In addition to geometric series, common series of numbers include the harmonic series, the alternating harmonic series, and p-series.
• Enduring understanding LIM-7: Applying limits may allow us to determine the finite sum of infinitely many terms.

• LIM-7.A Determine whether a series converges or diverges.
• LIM-7.A.8 The comparison test is a method to determine whether a series converges or diverges.
• LIM-7.A.9 The limit comparison test is a method to determine whether a series converges or diverges.
• Enduring understanding LIM-7: Applying limits may allow us to determine the finite sum of infinitely many terms.

• LIM-7.A Determine whether a series converges or diverges.
• LIM-7.A.10 The alternating series test is a method to determine whether an alternating series converges.
• Enduring understanding LIM-7: Applying limits may allow us to determine the finite sum of infinitely many terms.

• LIM-7.A Determine whether a series converges or diverges.
• LIM-7.A.11 The ratio test is a method to determine whether a series of numbers converges or diverges.
  • Exclusion statement: The nth term test for divergence, and the integral test, comparison test, limit comparison test, alternating series test, and ratio test for convergence are assessed on the AP Calculus BC Exam. Other methods are not assessed on the exam. However, teachers may include additional methods in the course, if time permits.
• Enduring understanding LIM-7: Applying limits may allow us to determine the finite sum of infinitely many terms.

• LIM-7.A Determine whether a series converges or diverges.
• LIM-7.A.12 A series may be absolutely convergent, conditionally convergent, or divergent.
• LIM-7.A.13 If a series converges absolutely, then it converges.
• LIM-7.A.14 If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value.
• Enduring understanding LIM-7: Applying limits may allow us to determine the finite sum of infinitely many terms.

• LIM-7.B Approximate the sum of a series.
• LIM-7.B.1 If an alternating series converges by the alternating series test, then the alternating series error bound can be used to bound how far a partial sum is from the value of the infinite series.
• Enduring understanding LIM-7: Applying limits may allow us to determine the finite sum of infinitely many terms.

• LIM-8.A Represent a function at a point as a Taylor polynomial.
• LIM-8.A.1 The coefficient of the nth degree term in a Taylor polynomial for a function f centered at x = a is f⁽ⁿ⁾(a)/n!.
• LIM-8.A.2 In many cases, as the degree of a Taylor polynomial increases, the nth degree polynomial will approach the original function over some interval.
• Enduring understanding LIM-8: Power series allow us to represent associated functions on an appropriate interval.

• LIM-8.B Approximate function values using a Taylor polynomial.
• LIM-8.B.1 Taylor polynomials for a function f centered at x = a can be used to approximate function values of f near x = a.
• Enduring understanding LIM-8: Power series allow us to represent associated functions on an appropriate interval.

• LIM-8.C Determine the error bound associated with a Taylor polynomial approximation.
• LIM-8.C.1 The Lagrange error bound can be used to determine a maximum interval for the error of a Taylor polynomial approximation to a function.
• LIM-8.C.2 In some situations, the alternating series error bound can be used to bound the error of a Taylor polynomial approximation to the value of a function.
• Enduring understanding LIM-8: Power series allow us to represent associated functions on an appropriate interval.

• LIM-8.D Determine the radius of convergence and interval of convergence for a power series.
• LIM-8.D.1 A power series is a series of the form Σ(n=0 to ∞) aₙ(x − r)ⁿ, where n is a non-negative integer, {aₙ} is a sequence of real numbers, and r is a real number.
• LIM-8.D.2 If a power series converges, it either converges at a single point or has an interval of convergence.
• LIM-8.D.3 The ratio test can be used to determine the radius of convergence of a power series.
• LIM-8.D.4 The radius of convergence of a power series can be used to identify an open interval on which the series converges, but it is necessary to test both endpoints of the interval to determine the interval of convergence.
• LIM-8.D.5 If a power series has a positive radius of convergence, then the power series is the Taylor series of the function to which it converges over the open interval.
• LIM-8.D.6 The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the same as the radius of convergence of the original power series.
• Enduring understanding LIM-8: Power series allow us to represent associated functions on an appropriate interval.

• LIM-8.E Represent a function as a Taylor series or a Maclaurin series.
• LIM-8.E.1 A Taylor polynomial for f(x) is a partial sum of the Taylor series for f(x).
• Enduring understanding LIM-8: Power series allow us to represent associated functions on an appropriate interval.

• LIM-8.F Interpret Taylor series and Maclaurin series.
• LIM-8.F.1 The Maclaurin series for 1/(1 − x) is a geometric series.
• LIM-8.F.2 The Maclaurin series for sin x, cos x, and eˣ provides the foundation for constructing the Maclaurin series for other functions.
• Enduring understanding LIM-8: Power series allow us to represent associated functions on an appropriate interval.

• LIM-8.G Represent a given function as a power series.
• LIM-8.G.1 Using a known series, a power series for a given function can be derived using operations such as term-by-term differentiation or term-by-term integration, and by various methods (e.g., algebraic processes, substitutions, or using properties of geometric series).
• Enduring understanding LIM-8: Power series allow us to represent associated functions on an appropriate interval.