Calculating Sums of Infinite Series: A Guide for Nille

In summary, the conversation revolves around finding the sum of infinite series, particularly the method for finding the sum of a geometric series and other series such as \sum_{n=0}^\infty\left(\frac{n+5}{5n+1}\right)^n and \sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}. The conversation also touches on the concepts of convergence and divergence, and the use of Taylor series to find the sum.
  • #1
nille40
34
0
Hi all!
I was wondering which method one should use to find the actual sum of an infinite series. I know how to find the sum of a geometric series (if it converges), but how could I find the sum for, for instance
[tex]
\sum_{n=0}^\infty\left(\frac{n+5}{5n+1}\right)^n
[/tex]

I know that it converges, and the sum appears to be 2. But how can I calculate this?

Or how about
[tex]
\sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}
[/tex]

Thanks in advance,
Nille
 
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  • #2
There is no general method for determining the sum of an infinite series.
 
  • #3
What if we try to find the sum to n terms and then taking lim [tex]n\rightarrow\infty[/tex]
 
  • #4
The answer to the first one is 2.5746952396343726343 Hope that will help
 
Last edited:
  • #5
The second one is a defined convergent series:

[tex]\sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}=\frac12\left(1+\pi{\rm csch}(\pi)\right)[/tex]

where csch(z) gives the hyperbolic cosecant of z, or in other words: csch(z)=1/sinh(z).
 
Last edited:
  • #6
How You guys reach this conclusions I have read a little about convergence and divergence but don't know how you summed up the series
 
  • #7
Originally posted by nille40
[tex]
\sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}
[/tex]

This one does not involve hyperbolic trig functions.

Taylor series.
 
  • #8
laura: would you mind explaining further? Since the sum, as written, is clearly not a Taylor series, do you mean that it can be converted to one and then summed? If so, how? It's certainly not obvious to me!
 

1. What is an infinite series?

An infinite series is a sum of an infinite number of terms. Each term in the series is added together to form the sum.

2. How do you calculate the sum of an infinite series?

To calculate the sum of an infinite series, you can use various methods such as the geometric series formula, the telescoping series method, or the ratio test. It is important to understand the convergence and divergence of the series before attempting to calculate the sum.

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

A convergent series is one in which the sum of the terms approaches a finite number as the number of terms increases. In contrast, a divergent series is one in which the sum of the terms does not approach a finite number and either increases or decreases without bound.

4. How do you determine the convergence or divergence of an infinite series?

To determine the convergence or divergence of an infinite series, you can use tests such as the limit comparison test, the integral test, or the p-series test. These tests compare the given series to a known convergent or divergent series and can help determine the behavior of the given series.

5. Can an infinite series have both positive and negative terms?

Yes, an infinite series can have both positive and negative terms. The sum of the series will depend on the behavior of the terms and whether they eventually cancel each other out or continue to add to the sum. It is important to consider the behavior of the terms when determining the convergence or divergence of the series.

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