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### series summation?

Posted: Mon Jan 03, 2011 5:47 am
Dear All
Can you please let me know how to calculate the summation when we know the series is convergent ? In other words, the test says that the series is convergent so how to calculate the resulted summation?
Thank you

### Re: series summation?

Posted: Mon Jan 03, 2011 6:26 am
That is a non-trivial problem. Geometric series are the only ones I know an easy formula for. Other innocent-looking series require much harder tools already. For example, you can show that 1+ 1/4 + 1/9 + 1/16 + ... converges to pi^2 / 6 using Fourier series or the residue theorem from complex analysis. Sometimes you can express a series in terms of other known series whose limit you know. However, most of the time the best you can do is find upper and lower bounds for the limit.

### Re: series summation?

Posted: Mon Jan 03, 2011 6:49 am
Thank you for your reply. Is there any straightforward manner to find the upper and lower band limits?

### Re: series summation?

Posted: Mon Jan 03, 2011 12:57 pm
Have you taken a calculus class before? The estimates typically used are those for the alternating series test, integral test, and of course the comparison test. On every GRE, they will ask you to evaluate a series, and you just need to play around with it and use obvious identities from freshman calculus to figure it out.

### Re: series summation?

Posted: Tue Jan 04, 2011 7:22 pm
hadimotamedi wrote:Thank you for your reply. Is there any straightforward manner to find the upper and lower band limits?

There can be many 'upper bounds'. For instance, in the series 1 + 1/2 + 1/4 + 1/8 + ... + 1/2^n + ... = 2 certainly the partial sums 1 + 1/2 + ... + 1/2^n are never greater than 3, or greater than 2.5, etc. (But if X < 2 then there are infinitely many n for which 1 + 1/2 + ... + 1/2^n is greater than X, so X is not a lower bound.)

What you want in this case is the least upper bound of the sequence of partial sums, because the sequence is monotone increasing.

The term 'least upper bound' should be familiar to you from real analysis. If it is not, then you should consult a good textbook (Walter Rudin's Principles of Mathematical Analysis, for instance) because there are many other associated concepts that depend on it.