Extra Problems for Chapter 9

Determine whether the sequences 1-5 converge and find the limits when they exist.

Determine whether the series 6-12 converge and find the sums when they exist.

Test the series 13-23 for convergence.

Test the series 27-30 by the Ratio Test.

Find the radius of convergence of the power series in Problems 31-35.

36            Find the interval of convergence of

37            Find the power series and radius of convergence for f'(x) and where

38            Find a power series for f(x) = 1/(1 + 2x³) and determine its radius of convergence.

39            Find a power series for

and determine its radius of convergence.

40            Approximate within 0.0001.

41            Approximate within 0.001.

42            Approximate

43            Approximate within 0.01.

44            Find a power series for (1 + x³)^-3/2 and give its radius of convergence.

45            Find a power series for and determine its radius of convergence.

46            Prove that any repeating decimal 0.b₁b₂...b_nb₁b₂...b_nb₁b₂...b_n... (where each of b₁, ..., b_n is a digit from the set {0,1,..., 9}) is equal to a rational number.

47            Approximately how many terms of the harmonic series 1 + ½ + ⅓ + ... + l/n + ... are needed to reach a partial sum of at least 50?

Hint: Compare with

48            Suppose and is either finite or ∞. Prove that

49            Suppose is a convergent positive term series and is a rearrangement of

. Prove that converges and has the same sum. Hint: Show that each

finite partial sum of is less than or equal to each infinite partial sum of ,

and vice versa.

50            Give a rearrangement of the series 1 - ½ + ⅓ - ¼ + ... which diverges to - x.

51             Suppose , and a_n≤ c_n≤ b_n for all n. Prove that

52            Prove the following result using the Limit Comparison Test.

Let and be positive term series and suppose lim_n→∞ (a_n/b_n) exists. If

converges then converges. If diverges then diverges.

53            Multiplication of Power Series. Prove that if f(x) =and g(x) =then f(x)g(x) =where c_n = a₀b_n + a₁ b_n-1 + ... + a_n-1b₁ + a_nb₀.

Hint: First prove the corresponding formula for partial sums, then take the standard part of an infinite partial sum.

54            Suppose f(x) is the sum of a power series for |x| < r and let g(x) = f(x²). Prove that for each n,

55             Show that if p ≤ -1 then the binomial series

diverges at x = 1 and x = -1. Hint: Cauchy Test.

If p ≥ 1, the series converges at x = 1 and x = -1. Hint: Compare with Note: The cases -1 < p < 1 are more difficult. It turns out that if -1 < p < 0 the series converges at x = 1 and diverges at x= -1. If p ≥ 0 the series converges at x = 1 and x = -1.

56             Prove that e is irrational, that is, e ≠ a/b for all integers a, b.

Hint: Suppose e = a/b, e^-1 = b/a. Let . Then |c| ≥ 1/a! but |c| ≤ l/(a +1)!.