Does the Integral Test Result Indicate the Series Sum?

In summary, the conversation discusses using the integral test to find the sum of a series and whether the result of the integral represents the sum or something else. It is mentioned that the result of the integral can serve as a first approximation of the sum, but other integrals may be used for better approximations.
  • #1
rick906
12
0
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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  • #2
No it does not.
 
  • #3
Thanks for the fast reply dude!

If that number is not the sum, does it represent something?
(just outta curiosity)
Thank you
 
Last edited:
  • #4
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
 

FAQ: Does the Integral Test Result Indicate the Series Sum?

What is the Integral Test?

The Integral Test is a method used in calculus to determine whether an infinite series converges or diverges. It involves comparing the series to a related improper integral.

How do you use the Integral Test?

To use the Integral Test, you first need to check that the series is positive and decreasing. Then, you set up a related improper integral with the same terms as the series. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

Can the Integral Test be used for all series?

No, the Integral Test can only be used for series with positive, continuous, and decreasing terms.

What is the difference between a convergent series and a divergent series?

A convergent series is one in which the terms of the series approach a finite limit as the number of terms increases. A divergent series is one in which the terms of the series do not approach a finite limit, but instead increase or decrease without bound.

Are there any limitations to the Integral Test?

Yes, the Integral Test can only be used to determine convergence or divergence, it cannot provide an exact sum for a convergent series. Additionally, the Integral Test may not work for more complex series and may require other methods such as the Comparison Test or the Ratio Test.

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