Chapter 11

(a) Find the Taylor polynomials up to degree 5 for \( f (x) = sin x \) centered at \( a = 0. \) Graph \( f \) and these polynomials on a common screen. (b) Evaluate \( f \) and these polynomials at \( x = \pi/4, \pi/2, \) and \( \pi \). (c) Comment on how the Taylor polynomials converge to \( f(x). \)

If the radius of convergence of the power series \( \sum_{n = 0}^{\infty} c_n x^n \) is 10, what is the radius of convergence of the series \( \sum_{n = 1}^{\infty} nc_n x^{n -1}? \) Why?

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^2 - 1}{n^3 + 1} \)

What can you say about the series \( \sum a_n \) in each of the following cases? (a) \( \displaystyle \lim_{n \to \infty} \mid \frac {a_{n + 1}}{a_n} \mid = 8 \) (b) \( \displaystyle \lim_{n \to \infty} \mid \frac {a_{n + 1}}{a_n} \mid = 0.8 \) (c) \( \displaystyle \lim_{n \to \infty} \mid \frac {a_{n + 1}}{a_n} \mid = 1 \)

(a) What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can you say about the remainder after \( n \) terms?

Suppose \( \sum a_n \) and \( \sum b_n \) are series with positive terms and \( \sum b_n \) is known to be convergent. (a) If \( a_n > b_n \) for all \( n, \) what can you say about \( \sum a_n? \) Why? (b) If \( a_n < b_n \) for all \( n, \) what can you say about \( \sum a_n? \) Why?

1\. Draw a picture to show that \( \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n^{1.3}} < \int^{\infty}_1 \frac {1}{x^{1.3}} dx \) What can you conclude about the series?

(a) What is the difference between a sequence and a series? (b) What is a convergent series? What is a divergent series?

(a) What is a sequence? (b) What does it mean to say that \( \lim_{n \to \infty} a_n = 8? \) (c) What does it mean to say that \( \lim_{n \to \infty} a_n = \infty? \)

What is a power series?

(a) Find the Taylor polynomials up to degree 3 for \( f(x) = tan x \) center at \( a = 0. \) Graph \( f \) and these polynomials on a common screen. (b) Evaluate \( f \) and these polynomials at \( x = \pi/6, \pi/4, \) and \( f(x). \) (c) Comment on how the Taylor polynomials converge to \( f(x). \)

Suppose you know that the series \( \sum_{n = 0}^{\infty} b_n x^n \) converges for \( \mid x \mid < 2\. \) What can you say about the following series? Why? \( \sum_{n = 0}^{\infty} \frac {b_n}{n + 1} x^{n + 1} \)

(a) What is the radius of convergence of a power series? How do you find it? (b) What is the interval of convergence of a power series? How do you find it?

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n - 1}{n^3 + 1} \)

Determine whether the series is absolutely convergent or conditionally convergent. \( \displaystyle \sum_ {n = 1}^{\infty} \frac {( - 1)^{n-1)}}{\sqrt{n}} \)

Test the series for convergence or divergence. \( \frac {2}{3} - \frac {2}{5} + \frac {2}{7} - \frac { 2}{9} + \frac {2}{11} - \cdot \cdot \cdot \)

Suppose \( \sum a_n \) and \( \sum b_n \) are series with positive terms and \( \sum b_n \) is known to be divergent. (a) If \( a_n > b_n \) for all \( n, \) what can you say about \( \sum a_n? \) Why? (b) If \( a_n < b_n \) for all \( n, \) what can you say about \( \sum a_n? \) Why?

Suppose \( f \) is a continuous positive decreasing function for \( x \ge 1 \) and \( a_n = f(n). \) By drawing a picture, rank the following three quantities in increasing order. \( \int^6_1 f(x) dx \displaystyle \sum_{i = 1}^{5} a_i \displaystyle \sum_{i = 2}^6 a_i \)

Explain what it means to say that \( \sum_{n = 1}^{\infty} a_n = 5. \)

(a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = e^x, \) \( a = 1 \)

If \( f^{(n)} (0) = (n + 1)! \) for \( n = 0, 1, 2, . . . , \) find the Maclaurin series for \( f \) and its radius of convergence.

Find a power series representation for the function and determine the interval of convergence. \( f(x) = \frac {1}{1 + x} \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} ( - 1)^n nx^n \)

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} ( - 1)^n \frac {n^2 - 1}{n^3 + 1} \)

Determine whether the series is absolutely convergent or conditionally convergent. \( \displaystyle \sum_{n = 0}^{\infty} \frac {( - 1)^n}{5n + 1} \)

Test the series for convergence or divergence. \( -\frac {2}{5} + \frac {4}{6} - \frac {6}{7} + \frac {8}{8} - \frac {10}{9} + \cdot \cdot \cdot \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^3 + 8} \)

Use the Integral Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} n^{-3} \)

Calculate the sum of the series \( \sum_{n = 1}^{\infty} a_n \) whose partial sums are given. \( s_n = 2 - 3(0.8)^n \)

List the first five terms of the sequence. \( a_n = \frac {2^n}{2n + 1} \)

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = \sin x, \) \( a = \pi/6 \)

Find the Taylor series for \( f \) centered at 4 if \( f^{(n)} (4) = \frac {(-1)^n n!}{3^n (n + 1)} \) What is the radius of convergence of the Taylor series?

Find a power series representation for the function and determine the interval of convergence. \( f(x) = \frac {5}{1 - 4x^2} \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {( - 1)^n x^n}{\sqrt[3]{n}} \)

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} ( -1)^n \frac {n^2 - 1}{n^2 + 1} \)

Determine whether the series is absolutely convergent or conditionally convergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n}{n^3 + 1} \)

Test the series for convergence or divergence. \( \frac {1}{\ln 3} - \frac {1}{\ln 4} + \frac {1}{\ln 5} - \frac {1}{\ln 6} + \frac {1}{\ln 7} - \cdot \cdot \cdot \)

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 2}^{\infty} \frac {1}{\sqrt n - 1} \)

Use the Integral Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} n^{-0.3} \)

Calculate the sum of the series \( \sum_{n = 1}^{\infty} a_n \) whose partial sums are given. \( s_n = \frac {n^2 - 1}{4n^2 + 1} \)

.List the first five terms of the sequence. \( a_n = \frac {n^2 - 1}{n^2 + 1} \)

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = \cos x, \) \( a = \pi/2 \)

Use the definition of a Taylor series to find the first four nonzero terms of the series for \( f(x) \) centered at the given value of \( a. \) \( f(x) = xe^x, \) \( a = 0 % \)

Find a power series representation for the function and determine the interval of convergence. \( f(x) = \frac {2}{3 - x} \)

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {x^n}{2n - 1} \)

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \frac {e^n}{n^2} \)

Determine whether the series is absolutely convergent or conditionally convergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {\sin n}{n^2} \)

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{n+1}{n \sqrt{n}}\)

Use the Integral Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {2}{5n - 1} \)