How do you find partial sums of infinite series?

Jul 16, 2015

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: $\setminus {\sum}_{k = 1}^{\setminus \infty} {a}_{k} = {a}_{1} + {a}_{2} + {a}_{3} + {a}_{4} + \setminus \cdots$ (this represents a "pretending" to add infinitely many numbers...even though such a thing is technically impossible to do). The infinite sequence $\left({a}_{k}\right)$, also written as $\setminus \left\{{a}_{k} \setminus\right\}$ or ${a}_{1} , {a}_{2} , {a}_{3} , \setminus \ldots$, is called the sequence of terms of the infinite series.

If, for each positive integer $n$, we let ${s}_{n}$ be defined as the finite sum ${s}_{n} = \setminus {\sum}_{k = 1}^{n} {a}_{k} = {a}_{1} + {a}_{2} + {a}_{3} + \setminus \cdots + {a}_{n}$, then the sequence $\left({s}_{n}\right)$, also written as $\setminus \left\{{s}_{n} \setminus\right\}$ or ${s}_{1} , {s}_{2} , {s}_{3} , \setminus \ldots$, is called the sequence of partial sums of the infinite series.

The two sequences $\left({a}_{k}\right)$ and $\left({s}_{n}\right)$ are related to each other, but they are different, and it's important to understand the relationships and differences.

It is the sequence $\left({s}_{n}\right)$ that determines whether the original series $\setminus {\sum}_{k = 1}^{\setminus \infty} {a}_{k}$ converges or not ("sums" to a finite number or not). In fact, by definition , if $\left({s}_{n}\right)$ converges to a number $s$, then the series $\setminus {\sum}_{k = 1}^{\setminus \infty} {a}_{k}$ is also said to converge to $s$ and we write that it "equals" $s$: $\setminus {\sum}_{k = 1}^{\setminus \infty} {a}_{k} = s$ ( 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 $\left({s}_{n}\right)$ does not converge, if it diverges (in any way), then the series $\setminus {\sum}_{k = 1}^{\setminus \infty} {a}_{k}$ is also said to diverge.

Here are two important things to realize with regard to the sequence of terms $\left({a}_{k}\right)$ and the sequence of partial sums $\left({s}_{n}\right)$:

1) If the sequence $\left({a}_{k}\right)$ does not converge to zero (meaning it either converges to some other number or it diverges), then the sequence $\left({s}_{n}\right)$ diverges, meaning that the series $\setminus {\sum}_{k = 1}^{\setminus \infty} {a}_{k}$ diverges.

2) If the sequence $\left({a}_{k}\right)$ does converge to zero, then the sequence $\left({s}_{n}\right)$ may or may not converge (so the series $\setminus {\sum}_{k = 1}^{\setminus \infty} {a}_{k}$ 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.