Real Analysis, Series, Tests.

Hi, I am running short on time, and I am having problems solving the following questions. I do not have any progress on them yet, but I am currently working on it. So any input is very welcome, even if it is just a basis advice on which direction to go with this stuff. Thanks for your time.

1. Find a series Σan which diverges by the Root Test but for which the Ratio Test gives no information.

2. Show that if Σan and Σbn are convergent series of nonnegative numbers, then Σ√anbn (square root of anbn) converges. Hint: Show that √anbn < or = an + bn for all n.

3. a. Prove that if Σ|an| converges and (bn) is a bounded sequence, then Σanbn converges. Hint: Use Theorem 14.4 (a series converges if and only if it satisfies the Cauchy criterion).

b. Observe that Corollary 14.7 (absolutely convergent series are convergent) is a special case of part (a)

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