How Can You Simplify the Process of Calculating Partial Sums?

[Stratosphere]

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?

 

[mathman]

You haven't defined what the terms in the sum are, so there is no way of knowing what can be done.

 

[Stratosphere]

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:

$$\sum^{\infty}_{n=0} \frac{(-1)^{n}x^{2n}}{n!}$$

 

[mathman]

For the particular example the sum is exp(-x²). For this case, there is no way to get partial sums except by direct addition.