(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]b_n[/itex] be a bounded sequence of nonnegative numbers. Let r be a number such that [itex]0 \leq r < 1[/itex].

Define [itex]s_n = b_1*r + b_2*r^2 + ... + b_n*r^n[/itex], for all natural numbers n.

Prove that [itex]{s_n}[/itex] converges.

2. Relevant equations

Sum of first n terms of geometric series = [itex]sum_n = (a_1)(1-r^{n+1})/(1-r)[/itex]

3. The attempt at a solution

Clearly, [itex]{s_n}[/itex] is monotonically increasing.

Since [itex]{b_n}[/itex] is bounded, [itex]|b_n| \leq M[/itex], for all natural numbers n.

I want to use the fact that if [itex]{s_n}[/itex] is both monotonically increasing and is bounded, then it must converge. The part of the problem that has stumped me for the past 45 minutes is how to show that [itex]{s_n}[/itex] is bounded.

The only material we've covered regarding infinite series thus far is for purely geometric series, which doesn't fit this problem precisely--but I included the formula anyway.

Help would be greatly appreciated! I have an exam on Monday and getting so completely stymied by a simple problem is not doing wonders for my confidence.

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# Convergence of a particular infinite sum

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