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I'm given $\{b_n\}$ is a bounded set of non-negative numbers and $r$ such that $0 \leq r < 1$. I need to show the sequence $\{s_n\} = b_1r +b_2r^2 + ... +b_nr^n $ converges. Ideally with the monotone convergence theorem.

So starting out I know $\{b_n\}$ is a bounded set. So $|s_n| \leq Mr + Mr^2 + ... + Mr^n$ and $Mr + Mr^2 + ... + Mr^n = M(r+r^2 + ... + r^n) = M(\frac{1}{1-r})$ so $|s_n| \leq M\frac{1}{1-r}$ giving me that $s_n$ is bounded. I also know that $r,r^2,r^3,...,r^n$ is decreasing monotonically since $0 \leq r < 1$

But at this point I get stuck. I don't know anything about $\{b_n\}$ except that it is bounded and non-negative. Can anybody give me a hint as to where I should be going from here? I know I need to show that somehow $\{s_n\}$ is monotone increasing since we keep adding to it in smaller and smaller amounts. But I don't know how I can show this with the information I have.

Any hints / pointers would be much appreciated.

Also I would ask that people refrain from posting a full proof. I'm trying to learn this material.

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+1 for the last sentence! – AD. Jan 20 '12 at 8:08
up vote 2 down vote accepted

You showed that the partial sums are bounded. By your hypotheses, they are increasing. This means they converge (do you know the theorem that a bounded monotonic sequence converges?).

You asked about showing that $s_n$ is increasing. Each term of the sequence is nonnegative (why?) so this should be clear. What don't you understand?

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Yes, it appears that doing proofs at midnight has its drawbacks. I see my error. I wasn't thinking of the sequence s_n as a sum but focusing on the individual terms. I figured out the rest of my proof now. – AvatarOfChronos Jan 20 '12 at 5:04

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