Lovely summation

by cjquines0, Oct 6, 2016, 11:54 AM

For those of you who haven’t checked out Chen's new summation handout, do check it out. This problem is a practice problem from said handout. He has two other new handouts as well, which are both really good.
Putnam 2011 wrote:
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1}a_n - 2$ for $n = 2, 3, \ldots$. Assume that the sequence $(b_j)$ is bounded. Prove that $$S = \sum_{n=1}^{\infty} \frac{1}{a_1 \cdots a_n}$$converges, and evaluate $S$.

Cleaned up solution

False starts

Tidbit

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Comment?

by shiningsunnyday, Oct 6, 2016, 12:16 PM

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Magic yessssssss. I play Magic.

by JunaBug13, Oct 6, 2016, 1:40 PM

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I also play magic!

by Kaladesh, Oct 6, 2016, 10:00 PM

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SSD: YES I KNOW I WAS STUPID OKAY. to be fair, it only took me like forty minutes

by cjquines0, Oct 7, 2016, 10:24 AM

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My sol

by shiningsunnyday, Oct 8, 2016, 2:08 PM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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