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What's the result of the series?

$\displaystyle\sum_{k=1}^{\infty}{\frac{k \cdot 2^k}{\binom{2k}{k}}}$

Numerical results suggest the answer is $\pi+3$

But how to prove it?

LLMs give proofs using Hypergeometric function or Beta function, it's hard for me to know whether they are valid proofs.

ntysdd
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  • From the duplicate:"This would then lead to $\displaystyle \sum_{n=1}^{\infty}\frac{n2^{n}}{\binom{2n}{n}}=\pi +3$ and many other forms just by using a general formula." – Dietrich Burde Jun 03 '25 at 15:09
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    If it‘s hard to know what is a proof, you won‘t be in a better position with the answers you may get here. – wasn't me Jun 03 '25 at 15:17
  • @wasn'tme I just don't believe the LLMs, I've been tricked by them too many times. One of them even told me the series diverge. – ntysdd Jun 03 '25 at 16:01
  • https://math.stackexchange.com/questions/4627830/what-is-the-name-of-this-sequence-that-approximates-pi – ntysdd Jun 03 '25 at 16:02
  • OEIS A180875 is related. – ntysdd Jun 03 '25 at 16:05
  • That's a known problem with LLMs, it's called "hallucinating". And they don't even need mushrooms for that. But there's some risk with NI (natural intelligence), too, and some of them take mushrooms. – wasn't me Jun 03 '25 at 16:53

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