Chapter 11

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {\sin(n \pi /6)}{1 + n\sqrt{n}} \)

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{3^n + 4^n} \)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{1 + (\frac {2}{3})^n} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {(-1)^{n + 1}n}{n + \sqrt n} \)

A uniformly charge disk has a radius \( R \) and surface charge density \( \sigma \) as in the figure. The electric potential \( V \) at a point \( P \) at a distance \( d \) along the perpendicular central axis of the disk is \( V = 2 \pi k_e \sigma \left( \sqrt {d^2 + R^2} - d \right) \) where \( k_e \) is a constant (called Coulomb's constant). Show that \( V \approx \frac {\pi k_e R^2 \sigma}{d} \) for large \( d \)

The Bessel function of order 1 is defined by \( J_1 (x) = \sum_{n = 0}^{\infty} \frac {(-1)^n x^{2n + 1}}{n! (n + 1)! 2^{2n + 1}} \) (a) Show that \( J_1 \) satisfies the differential equation \( x^2 J''_1 (x) + xJ'_1(x) + (x^2 - 1) J_1(x) = 0 \) (b) Show that \( J'_0 (x) = -J_1 (x). \)

If a surveyor measures differences in elevation when making plans for a highway across a desert, corrections must be made for the curvature of the earth. (a) If \( R \) is the radius of the earth and \( L \) is the length of the highway, show that the correction is \( C = R \sec (L/R) - R \) (b) Use a Taylor polynomial to show that \( C \approx \frac {L^2}{2R} + \frac {5L^4}{24R^3} \) (c) Compare the corrections given by the formulas in part (a) and (b) for a highway that is 100 km long. (Take the radius of the earth to be 6370 km.)

(a) Show that the function \( f(x) = \sum_{n = 0}^{\infty} \frac {x^n}{n!} \) is a solution of the differential equation \( f'(x) = f(x) \) (b) Show that \( f(x) = e^x. \)

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

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n \arctan n}{n^2} \)

The meaning of the decimal representation of a number \( 0.d_1d_2d_3. . . .\) (where the digit \( d_i \) is one of the numbers 0, 1, 2, . . . , 9) is that \( 0.d_1d_2d_3d_4 . . . = \frac {d_1}{10} + \frac {d_2}{10^2} + \frac {d_3}{10^3} + \frac {d_4}{10^4} + . . . \) Show that this series always converges.

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \ln \left( \frac {n^2 + 1}{2n^2 + 1} \right) \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( \left \\{ \frac {(2n - 1)!}{(2n + 1)!}\right \\}\)

A function \( f \) is defined by \( f(x) = 1 + 2x + x^2 + 2x^3 + x^4 + \cdot \cdot \cdot \) that is, its coefficients are \( c_{2n} = 1 \) and \( c_{2n + 1} = 2 \) for all \( n \ge 0. \) Find the interval of convergence of the series and find an explicit formula for \( f(x). \)

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. \( f(x) = e^{3x} - e^{2x} \)

Let \( f_n (x) = \left( \sin nx \right)/n^2. \) Show that the series \( \sum f_n(x) \) converges for all values of \( x \) but the series of derivatives \( \sum f'_n (x) \) diverges when \( x = 2n \pi, n \) an integer. For what values of \( x \) does the series \( \sum f''_n (x) \) converge?

If \( f(x) = \sum_{n = 0}^{\infty} c_n x^n, \) where \( c_{n + 4} = c_n \) for all \( n \ge 0, \) find the interval of convergence of the power series and a formula for \( f(x). \)

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

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 2}^{\infty} \frac {( - 1)^n}{n \ln n} \)

For what values of \( p \) does the series \( \sum_{n = 2}^{\infty} 1/(n^P \ln n) \) converge?

Find the sum of the series \( \sum_{n = 1}^{\infty} ne^{-2n} \) correct to four decimal places.

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$ \displaystyle \sum_{k = 0}^{\infty} (\sqrt 2)^{-k} $$

Determine whether the sequence converges or diverges. If it converges, find the limit. \( \left \\{ \frac {\ln n}{\ln 2n} \right \\} \)

Let \( f(x) = \sum_{n = 1}^{\infty} \frac {x^n}{n^2} \) Find the intervals of convergence for \( f, f', \) and \( f''. \)

Show that if \( \lim_{n \to \infty} \sqrt[n]{\mid c_n \mid} = c, \) where \( c \not= 0, \) then the radius of convergence of the power series \( \sum c_n x^n \) is \( R = 1/c. \)

The terms of a series are defined recursively by the equations \( a_1 = 2 \) \( a_{n+1} = \frac {5n + 1}{4n + 3} a_n \) Determine whether \( \sum a_n \) converges or diverges.

Prove that if \( a_n \ge 0 \) and \( \sum a_n \) converges, then \( \sum a_n^2 \) also converges.

Estimate \( \sum_{n = 1}^{\infty} (2n + 1)^{-6} \) correct to five decimal places.

In Section 4.8 we considered Newton's method for approximating a root \( r \) of the equation \( f(x) = 0, \) and from an initial approximation \( x_1 \) we obtained successive approximations \( x_2, x_3, . . . . , \) where \( x_{n + 1} = x_n - \frac {f (x_n)}{f' (x_n)} \) Use Taylor's Inequality with \( n = 1, a = x_n \) and \( x = r \) to show that if \( f''(x) \) exists on an interval \( I \) containing \( r, x_n, \) and \( x_{n + 1}, \) and \( \mid f''(x) \mid \le M, \mid f'(x) \mid \ge K \) for all \( x \in I, \) then \( \mid x_{n + 1} - r \mid \le \frac {M}{2K} \mid x_n - r \mid^2 \) [This means that if \( x_n \) is accurate to \( d \) decimal places, then \( x_{n + 1} \) is accurate to about \( 2d \) decimal places. More precisely, if the error at stage \( n \) is at most \( 10^{-m}, \) then the error at stage \( n + 1 \) is at most \( (M/2K)10^{-2m}. \)]

(a) Starting with the geometric series \( \sum_{n = 0}^{\infty} x^n, \) find the sum of the series \( \sum_{n = 1}^{\infty} nx^{n - 1} \mid x \mid < 1 \) (b) Find the sum of each of the following series. (i) \( \sum_{n = 1}^{\infty} nx^n, \mid x \mid < 1 \) (ii) \( \sum_{n = 1}^{\infty} \frac {n}{2^n} \) (c) Find the sum of each of the following series. (i) \( \sum_{n = 2}^{\infty} n(n - 1) x^n, \mid x \mid < 1 \) (ii) \( \sum_{n = 2}^{\infty} \frac {n^2 - n}{2^n} \) (iii) \( \sum_{n = 1}^{\infty} \frac {n^2}{2^n} \)

Suppose that the power series \( \sum c_n (x - a)^n \) satisfies \( c_n \not= 0 \) for all \( n. \) Show that if \( \lim_{n \to \infty} \mid c_n/c_{n + 1} \mid \) exists, then it is equal to the radius of convergence of the power series.

A series \( \sum a_n \) is defined by the equations \( a_1 = 1 \) \( a_{n+1} = \frac {2 + \cos n}{\sqrt{n}} a_n \) Determine whether \( \sum a_n \) converges or diverges.

(a) Suppose that \( \sum a_n \) and \( \sum b_n \) are series with positive terms and \( \sum b_n \) is convergent. Prove that if \( \displaystyle \lim_{n \to \infty} \frac {a_n}{b_n} = 0 \) then \( \sum a_n \) is also convergent. (b) Use part (a) to show that the series converges. (i) \( \displaystyle \sum_{n = 1}^{\infty} \frac {\ln n }{n^3} \) (ii) \( \displaystyle \sum_{n = 1}^{\infty} \frac { \ln n}{\sqrt n e^n} \)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac {3}{5^n} + \frac {2}{n} \right) \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {\tan^{-1}n}{n} \)

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. \( f(x) = x^2 \ln (1 + x^3) \)

How many terms of the series \( \sum_{n = 2}^{\infty} 1/[n(\ln n)^2] \) would you need to add to find its sum to within 0.01?

Use the power series for \( tan^{-1} x \) to prove the following expression for \( \pi \) as the sum of an infinite series: \( \pi = 2 \sqrt 3 \sum_{n = 0}^{\infty} \frac {(-1)^n}{(2n + 1) 3^n} \)

Suppose the series \( \sum c_n x^n \) has radius of convergence 2 and the series \( \sum d_n x^n \) has radius of convergence 3. What is the radius of convergence of the series \( \sum (c_n + d_n) x^n? \)

(a) Suppose that \( \sum a_n \) and \( \sum b_n \) are series with positive terms and \( \sum b_n \) is divergent. Prove that if \( \displaystyle \lim_{n \to \infty} \frac {a_n}{b_n} = \infty \) then \( \sum{a_n} \) is also divergent. (b) Use part (a) to show that the series diverges. (i) \( \displaystyle \sum_{n = 2}^{\infty} \frac {1}{\ln n } \) (ii) \( \displaystyle \sum_{n = 1}^{\infty} \frac {\ln n}{n} \)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac {1}{e^n} + \frac {1}{n(n + 1)} \right) \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( \\{ n^2e^{-n}\\} \)

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. \( f(x) = \frac {x}{\sqrt {4 + x^2}} \)

Let \( \\{ {b_n} \\} \) be a sequence of positive numbers that converges to \( \frac {1}{2}. \) Determine whether the given series is absolutely convergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {{{b_{n }^{n} \cos n \pi }}}{n} \)

Show that if we want to approximate the sum of the series \( \sum_{n = 1}^{\infty} n^{-1.001} \) so that the error is less than 5 in the ninth decimal place, then we need to add more than \( 10^{11,301} \) terms!

Suppose that the radius of convergence of the power series \( \sum c_n x^n \) is \( R. \) What is the radius of convergence of the power series \( \sum c_n x^{2n} ? \)

Let \( \\{ {b_n} \\} \) be a sequence of positive numbers that converges to \( \frac {1}{2}. \) Determine whether the given series is absolutely convergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n n!}{n^nb_1b_2b_3 \cdot \cdot \cdot b_n} \)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {e^n}{n^2} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \ln (n + 1) - \ln n \)

Give an example of a pair of series \( \sum a _n \) and \( \sum b_n \) with positive terms where \( \lim_{n \to \infty} (a_n/b_n) = 0 \) and \( \sum b_n \) diverges, but \( \sum a_n \) converges. (Compare with Exercise 40.)