# How do you find partial sums of infinite series?

##### 1 Answer

The sequence of partial sums of an infinite series is a sequence created by taking, in order: 1) the first term, 2) the sum of the first two terms, 3) the sum of the first three terms, etc..., forever and ever.

#### Explanation:

An infinite series is an expression of the form: **sequence of terms** of the infinite series.

If, for each positive integer *finite* sum **sequence of partial sums** of the infinite series.

The two sequences *related to each other, but they are different*, and it's important to understand the relationships and differences.

It is the sequence *definition* , if *understand that this is just convenient notation...we are not literally adding up infinitely many numbers, though it usually doesn't hurt to pretend we are* ).

On the other hand, if

Here are two important things to realize with regard to the sequence of terms

1) If the sequence

2) If the sequence *may or may not converge* (so the series *may or may not converge* ). We need *other tests* to help us decide about convergence (a formula for the partial sums (if possible), ratio test, root test, comparison test, limit comparison test, alternating series test, absolute convergence test, etc...)

Since this answer is so long, I'll let you look in your textbook or online for examples to illustrate these ideas. You might start by using the following search terms: "geometric series", "p series", "harmonic series", "alternating harmonic series", as well as the tests mentioned in the previous paragraph.