Does the Integral Test Result Indicate the Series Sum?

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Discussion Overview

The discussion revolves around the Integral Test in the context of series convergence, specifically questioning whether the result of the integral indicates the actual sum of the series. Participants explore the implications of the integral test results and their interpretations.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions whether the result of the integral test, which indicates convergence, also signifies that the series converges to the value of the integral.
  • Another participant asserts that the result of the integral does not represent the sum of the series.
  • A follow-up inquiry seeks to understand if the integral result signifies something else if it is not the sum, expressing curiosity about its meaning.
  • A later reply suggests that the integral result can serve as a first approximation of the sum, providing an example involving the logarithm of factorials and its relation to integrals for better approximations.

Areas of Agreement / Disagreement

There is disagreement regarding whether the integral result indicates the actual sum of the series. Some participants assert it does not, while others propose it may serve as an approximation.

Contextual Notes

Participants do not fully explore the conditions under which the integral test applies or the specific nature of the approximations mentioned, leaving some assumptions and definitions unaddressed.

rick906
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Hi all,
I just want to know a little something:
When doing the integral test in order to find a sum, when might get a result (integral) of a certain number. As we know, getting a number as result an integral test means that this serie converges...but does that mean that the serie converges to this (the result of the integral...number we just found)?
I don't think so, but I'm not sure either.

Thanks for the info
 
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No it does not.
 
Thanks for the fast reply dude!

If that number is not the sum, does it represent something?
(just outta curiosity)
Thank you
 
Last edited:
rick906 said:
If that number is not the sum, does it represent something?
(just outta curiosity)
It is a first approximations of the sum.
For example say we desired to know
log(n!)=sum[log(k),{k,1,n}]~Integral[log(k),{k,0,n}]~n*log(n)-n
other integrals can be used for better approximations
 

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