conceptual summation problem

**flipperpk** · November 3rd 2009, 10:42 AM

Let $\sum_{k=1}^{\infty}a_n$ be a convergent series.

Let $|\frac{a_{n+1}}{a_n}| < r < 1$ , $n \in N$ .

Let $(b_n)$ be a sequence such that $|b_n|< M$ .

Prove that $\sum_{k=1}^{\infty}a_n b_n$ is convergent.

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I see here that

$|\frac{a_{n+1}}{a_n}| < r < 1$ looks like the ratio test.

$(b_n)$ has upperbound M.

How do I go about this proof?

**Jameson** · November 3rd 2009, 11:03 AM

There is already a proof about this situation:

Abel's test - Wikipedia, the free encyclopedia

The only thing lacking is b_n being monotonic, but somehow I think that's assumed.

**girdav** · November 3rd 2009, 11:43 AM

We have $\left|c_{n+1}\right| =\left|a_{n+1}b_{n+1}\right| \leq rM\left|a_n\right| \leq \cdots \leq r^{n+1}.M$

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