Thread: conceptual summation problem

1. conceptual summation problem

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.

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