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Homework 11 Math 501 Due November 21, 2014 Exercise 1 Let (an ) ⊂ R be a sequence such that an ≥ 0 for all n and Prove that ∞ √ X an n n=1 P∞ n=1 an converges. converges. [Hint: First show that if x, y > 0 then xy ≤ 12 (x2 + y 2 ).] Exercise 2 Prove that the series ∞ X 1 p n log n n=1 converges when p > 1 and diverges when p ≤ 1. [Note: logp n simply means (log n)p in the same way sin2 x means (sin x)2 . Moreover, log means the natural logarithm with base e.] Exercise 3 Prove the positive part of the limit comparison test. That is, suppose that (an ), (bn ) ⊂ R are sequences of positive real numbers and an < ∞. bn P∞ P∞ Prove that if n=1 bn converges, then n=1 an converges. lim sup n→∞ Exercise 4 P∞ Let (an ) ⊂ R be a sequence of a positive real numbers. Prove that n=1 an P∞ converges if and only if n=1 log(1 + an ) converges. [Hint: use L’Hò‚pital’s rule and the limit comparison test.] 1 Exercise 5 Q∞ An infinite product is a formal product n=1 cn whose factors are positive real numbers. The nth partial product is Cn = c1 · · · cn . If Cn converges to a limit C 6= 0, then the infinite product is said to converge to C. Q∞Suppose cn = 1 + an where anP≥∞0 for all n or an ≤ 0 for all n. Prove that n=1 cn converges if and only if n=1 an converges. Exercise 6 P∞ Let (an ), (bn ) ⊂ R be sequences of real numbers. P∞ Prove that if n=1 |an | converges and (bn ) is a bounded sequence, then n=1 an bn converges. Exercise 7 Let fn , f : [a, b] → R be functions. Suppose that fn → f pointwise and Z b Z fn dx → a f dx a as n → ∞. Prove or disprove: fn ⇒ f . 2 b