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Let $\sum _{n=1}^{\infty} a_{n} $and $\sum _{n=1}^{\infty} b_{n} $ converge absolutely. Prove that $\sum_{n=1}^{\infty} \sqrt{|a_{n}b_{n}|} $ converges.

I know that the series $\sum_{n=1}^{\infty} a_{n}b_{n}$ converges absolutely, but am having trouble showing what they want. I have tried showing the partial sums are bounded but no luck so far.

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4 Answers 4

up vote 12 down vote accepted

Hint: $\sqrt{|a_nb_n|}\leq (|a_n|+|b_n|)/2$.

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Are you familiar with the Cauchy-Schwarz inequality? $$ \sum x_i y_i \leq \left(\sum x_i^2 \right)^{1/2} \left(\sum y_i^2\right)^{1/2}. $$

In this case take $x_i = \sqrt{|a_i|}, y_i = \sqrt{|b_i|}$ and we find $$ \sum \sqrt{|a_n b_n|} \leq \left(\sum |a_i| \right)^{1/2} \left(\sum|b_i|\right)^{1/2} < \infty. $$

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Let $\sum _{n=1}^{\infty} a_{n} $and $\sum _{n=1}^{\infty} b_{n} $ converge absolutely. Prove that $\sum_{n=1}^{\infty} \sqrt{|a_{n}b_{n}|} $ converges.

Since $\sum |a_n|$ and $\sum |b_n|$ converge, $\;\sum (|a_n| + |b_n|)\;$ converges.

Then, essentially, you need only to prove that $$\sqrt{|a_n b_n|} \leq \frac{1}{2}(|a_n| + |b_n|).$$

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+1 Hello, :-) I made this answer favorite , Amy –  B. S. Aug 29 '13 at 0:51
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Hint: 1) Note that $\sum |a_n|$ and $\sum |b_n|$ converge.

2) Prove that $\sum (|a_n| + |b_n|)$ converges.

3) Prove that $\sqrt{|a_n b_n|} \leq \frac{1}{2}(|a_n| + |b_n|)$, thus $\sum\sqrt{|a_n b_n|}$ converges.

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