Convergence of Infinite Series with Increasing Denominators

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Discussion Overview

The discussion centers around the convergence of an infinite series with increasing denominators, specifically the series S=1/2+1/(2*4)+1/(2*4*6)... Participants explore whether the series converges and how to calculate its sum, touching on concepts related to Taylor series.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant expresses doubt about the convergence of the series, suggesting it does not converge.
  • Another participant asserts that the series does converge and identifies it as a well-known series related to the Taylor series of e^x.
  • A later reply seeks clarification on how to calculate the sum of the series, indicating a need for further understanding of the mathematical concepts involved.
  • One participant claims to have found the sum independently, referencing the Taylor expansion of e^x at 1/2.

Areas of Agreement / Disagreement

There is disagreement regarding the convergence of the series, with some participants asserting it converges while others express uncertainty. The discussion does not reach a consensus on the method of calculation.

Contextual Notes

Participants reference Taylor series and mathematical concepts without fully resolving the steps or assumptions involved in the convergence and calculation of the series.

pixel01
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S=1/2+1/(2*4)+1/(2*4*6)...+1/(2*4*6..2n)+...

I want to calculate the sum of this series, but it seems not to converge.
Can anyone help me

Thank you.
 
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It does converge, the sum is[tex]\sum_{n=1}^{\infty} \frac{1}{n!2^n}[/tex]

which is a well known series (what Taylor series do you know?).
 
Thank you, yes it does converge. Could you please tell me how to calculate the sum.
I should revise some old maths books.
 
He did tell you how to calculate the sum:
matt grime said:
which is a well known series (what Taylor series do you know?).
 
I found it myself. The Taylor expansion of e^x at 1/2. Thanks.
 

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