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.

 

[CellCoree]

I would recommend using mathematical techniques such as integration or series convergence tests to find the exact value of the sum. These methods can help simplify the process and make it easier to obtain the partial sum. Additionally, using technology such as a calculator or computer program can also aid in finding the exact value of the sum. It is important to carefully consider the limitations and assumptions of these methods in order to ensure the accuracy of the results.

 

[pmacias]

There are a few techniques that can make finding the partial sum easier and more efficient. One approach is to use mathematical properties such as commutativity and associativity to rearrange the terms in a way that allows for easier addition. Another technique is to use known formulas or patterns to simplify the terms before adding them up. Additionally, there are computational tools such as calculators or computer programs that can help with the calculation of the partial sum.

To find the exact value of the sum, we can use various methods such as the geometric series formula, telescoping series, or the method of differences. These methods involve manipulating the series and using known mathematical concepts to find the exact value of the sum. The choice of method may depend on the specific series and its properties. It is also important to check for convergence or divergence of the series before attempting to find the exact value.