The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Extra Problems for Chapter 9

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

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

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Test the series 13-23 for convergence.

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Test the series 27-30 by the Ratio Test.

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

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36            Find the interval of convergence of 09_infinite_series-841.gif

37            Find the power series and radius of convergence for f'(x) and 09_infinite_series-842.gifwhere

09_infinite_series-843.gif

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

39            Find a power series for

09_infinite_series-844.gif

and determine its radius of convergence.

40            Approximate 09_infinite_series-845.gif within 0.0001.

41            Approximate 09_infinite_series-846.gif within 0.001.

42            Approximate 09_infinite_series-847.gif

43            Approximate 09_infinite_series-848.gifwithin 0.01.

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

45            Find a power series for 09_infinite_series-849.gif and determine its radius of convergence.

46            Prove that any repeating decimal 0.b1b2...bnb1b2...bnb1b2...bn... (where each of b1, ..., bn 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 09_infinite_series-850.gif

48            Suppose 09_infinite_series-851.gif and 09_infinite_series-852.gif is either finite or ∞. Prove that 09_infinite_series-853.gif

49            Suppose 09_infinite_series-854.gif is a convergent positive term series and 09_infinite_series-855.gif is a rearrangement of

09_infinite_series-856.gif. Prove that 09_infinite_series-857.gif converges and has the same sum. Hint: Show that each

finite partial sum of 09_infinite_series-858.gif is less than or equal to each infinite partial sum of 09_infinite_series-859.gif,

and vice versa.

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

51             Suppose 09_infinite_series-860.gif, and an≤ cn≤ bn for all n. Prove that 09_infinite_series-861.gif

52            Prove the following result using the Limit Comparison Test.

Let 09_infinite_series-862.gifand 09_infinite_series-863.gif be positive term series and suppose limn→∞ (an/bn) exists. If

09_infinite_series-864.gif converges then 09_infinite_series-865.gif converges. If 09_infinite_series-866.gif diverges then 09_infinite_series-867.gif diverges.

53            Multiplication of Power Series. Prove that if f(x) =09_infinite_series-868.gifand g(x) =09_infinite_series-869.gifthen f(x)g(x) =09_infinite_series-870.gifwhere cn = a0bn + a1 bn-1 + ... + an-1b1 + anb0.

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,

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

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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 09_infinite_series-873.gif 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 09_infinite_series-874.gif. Then |c| ≥ 1/a! but |c| ≤ l/(a +1)!.


Last Update: 2006-11-25