Does the Integral Test Result Indicate the Series Sum?

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SUMMARY

The Integral Test is a method used to determine the convergence of a series, but the result of the integral does not equal the sum of the series. Instead, it provides a first approximation of the series sum. For instance, the approximation of log(n!) can be expressed as the integral of log(k) from 0 to n, yielding an approximation of n*log(n) - n. This indicates that while the integral confirms convergence, it does not provide the exact sum of the series.

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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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