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Extra Problems for Chapter 9
Determine whether the sequences 15 converge and find the limits when they exist. Determine whether the series 612 converge and find the sums when they exist. Test the series 1323 for convergence. Test the series 2730 by the Ratio Test. Find the radius of convergence of the power series in Problems 3135. 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 + 2x^{3}) 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 + x^{3})^{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.b_{1}b_{2}...b_{n}b_{1}b_{2}...b_{n}b_{1}b_{2}...b_{n}... (where each of b_{1}, ..., 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} = a_{0}b_{n} + a_{1} b_{n1} + ... + a_{n1}b_{1} + a_{n}b_{0}. 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(x^{2}). 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)!.


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