How Can You Simplify the Process of Calculating Partial Sums?

AI Thread Summary
Calculating partial sums can be complex, especially when dealing with infinite series. The discussion highlights that without defining the specific terms of the sum, it's challenging to determine an effective calculation method. In the provided example of the series involving exp(-x^2), direct addition is the only viable approach for obtaining partial sums. Participants express a desire for a general formula to simplify this process but acknowledge its absence in certain cases. Ultimately, the conversation underscores the need for clarity in defining series terms to explore potential simplifications.
Stratosphere
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Is there a particular way to get the partial sum easier than just adding the terms up?

In this formula it would take a while to add up the terms if I wanted to use n=20:

S_{n}+\int ^{\infty}_{n+1}f(x) dx\leqs\leq S_{n}+\int ^{\infty}_{n}f(x)dx

How would I get the exact value of the sum?
 
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You haven't defined what the terms in the sum are, so there is no way of knowing what can be done.
 
mathman said:
You haven't defined what the terms in the sum are, so there is no way of knowing what can be done.

Oh, I though that there was something like a formula that could be used in general cases. So I'll use the example:

\sum^{\infty}_{n=0} \frac{(-1)^{n}x^{2n}}{n!}
 
For the particular example the sum is exp(-x2). For this case, there is no way to get partial sums except by direct addition.
 
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