Sum of infinite series | Mathematics | Page 1

velvet fossil Jun 13, 2025, 1:04 PM

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$\frac{1}{2 * 4} + \frac{1 * 3}{2 * 4 * 6} + \frac{1 * 3 * 5}{2 * 4 * 6 * 8} + \text{... } \infty$

patent hollyBOT Jun 13, 2025, 1:04 PM

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Sypse

harsh geodeBOT Jun 13, 2025, 1:04 PM

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velvet fossil $\frac{1}{2 * 4} + \frac{1 * 3}{2 * 4 * 6} + \frac{1 * 3 * 5}{2 * 4 * 6 * 8} + \...

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velvet fossil Jun 13, 2025, 1:04 PM

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patent holly **Sypse**

I have to find the sum of the following, it does converge, and I do think it can be converted into a telescopic series

hidden trellis Jun 13, 2025, 1:13 PM

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patent holly **Sypse**

Watch your notation there with the infinity

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That implies it diverges

glacial vale Jun 13, 2025, 1:15 PM

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Well, the general term is 2^(-2n - 3) (2n + 2)!/((n + 1)! (n + 2)!).
And since (2n + 2)!/((n + 1)! (n + 2)!) is the (n + 1)th Catalan number, I believe you can express it as a difference of binomial coefficients. Though, that doesn't immediately make it telescopic.

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Catalan numbers have a lot of properties, so I'm sure there's a sum like that among them.

hidden trellis Jun 13, 2025, 1:16 PM

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This is what I got

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Doesn't look that nice tbf

glacial vale Jun 13, 2025, 1:17 PM

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You can bring it to factorials with some manipulation.

velvet fossil Jun 13, 2025, 1:18 PM

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hidden trellis Watch your notation there with the infinity

uh thats for jst showing there are infinite terms

hidden trellis Jun 13, 2025, 1:18 PM

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Hmm, is there a form of double factorials for Catalan numbers?

hidden trellis Jun 13, 2025, 1:18 PM

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velvet fossil uh thats for jst showing there are infinite terms

Just use $\ldots$ then

patent hollyBOT Jun 13, 2025, 1:18 PM

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Kocher

velvet fossil Jun 13, 2025, 1:18 PM

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ok 👍

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also as mentioned this can be broken down into a telescopic series via some manipulations

hidden trellis Jun 13, 2025, 1:20 PM

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So you've done it? Or is it speculation?

velvet fossil Jun 13, 2025, 1:20 PM

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from my teacher.

hidden trellis Jun 13, 2025, 1:21 PM

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$\sum_{n=2}^{\infty}\frac{(2n-3)!!}{(2n)!!}$

patent hollyBOT Jun 13, 2025, 1:21 PM

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Kocher

hidden trellis Jun 13, 2025, 1:21 PM

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Hmm, now this looks doable

glacial vale Jun 13, 2025, 1:21 PM

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hidden trellis Hmm, is there a form of double factorials for Catalan numbers?

Well, C(n) = (2n)!/(n! (n + 1)!), (2n - 1)!! = (2n)!/(2^n n!), so:
(2n)!/n! = 2^n (2n - 1)!!
C(n) = 2^n (2n - 1)!!/(n + 1)!
If you want it described that way.

hidden trellis Jun 13, 2025, 1:23 PM

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I'm thinking there might actually be a telescopic here

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What terms would you need to cancel out though thonk

velvet fossil Jun 13, 2025, 1:25 PM

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also the options are 1/2, 1/4, 1/3 and 1

glacial vale Jun 13, 2025, 1:26 PM

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Well, that doesn't matter.

glacial vale Jun 13, 2025, 1:26 PM

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hidden trellis $\sum_{n=2}^{\infty}\frac{(2n-3)!!}{(2n)!!}$

Ok, yeah, got this (just starting from n = 0):
2^(-2n - 3) (2n + 2)!/((n + 1)! (n + 2)!) = (2n + 2)!/((2n + 2)!! (2n + 4)!!) = (2n + 1)!!/(2n + 4)!!

hidden trellis Jun 13, 2025, 1:28 PM

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Okay as I'm doing this I'm getting more and more of a feeling that there is some telescope

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Perhaps let $a_{n+1}-a_n=\frac{(2n-3)!!}{(2n)!!}$

patent hollyBOT Jun 13, 2025, 1:29 PM

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Kocher

glacial vale Jun 13, 2025, 1:29 PM

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Yeah, that's what I'm trying to do, too.

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I'm going to assume some form of a(n).

velvet fossil Jun 13, 2025, 1:30 PM

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ok so I do know the answer now, the problem is I don't think I would be able to think it by myself.

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this can be written as $\frac{1}{2}(1-\frac{3}{4}) + \frac{13}{24}(1-\frac{5}{6}) + \ldots$

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$\implies \frac{1}{2} - \frac{13}{24} + \frac{13}{24} - \frac{135}{246} + \ldots$

patent hollyBOT Jun 13, 2025, 1:40 PM

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Sypse

hidden trellis Jun 13, 2025, 1:40 PM

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That is so random

patent hollyBOT Jun 13, 2025, 1:40 PM

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Sypse

velvet fossil Jun 13, 2025, 1:40 PM

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hidden trellis That is so random

🤷‍♂️

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thats what I am saying, how do I reach here by myself?

glacial vale Jun 13, 2025, 1:49 PM

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Well, here's one possible approach. This requires you to know the OGF of Catalan numbers, though.

glacial vale Jun 13, 2025, 1:49 PM

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velvet fossil $\implies \frac{1}{2} - \frac{1*3}{2*4} + \frac{1*3}{2*4} - \frac{1*3*5}{2*4*6} ...

Ohh, I see!

velvet fossil Jun 13, 2025, 1:50 PM

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glacial vale Well, here's one possible approach. This requires you to know the OGF of Catalan...

I do not. sad

glacial vale Jun 13, 2025, 1:51 PM

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velvet fossil thats what I am saying, how do I reach here by myself?

Well, it's not always very easy to notice telescoping.

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So, just by looking at the expression really hard, I guess 😅

velvet fossil Jun 13, 2025, 2:01 PM

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+close

light meteorBOT Jun 13, 2025, 2:01 PM

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velvet fossil +close

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light meteorBOT Jun 13, 2025, 2:02 PM

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light meteor

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#Sum of infinite series