# How do you find the sum of the infinite geometric series given a_1=14, r=7/3?

Oct 27, 2016

For a series to have a finite sum the general term of the series must tend to zero when $n \rightarrow \infty$

#### Explanation:

In other words, ${a}_{n} \rightarrow 0$ is a necessary condition for the series to be convergent, that is, to have a finite sum. (It is not sufficient though, see below).

But the general term of the given series is $14 {\left(\frac{7}{3}\right)}^{n}$, which does not tend to zero when $n$ tends to $\infty$. So the series doesn't have a finite sum, it is called divergent. It is often said that the series has an infinite sum.

A couple of remarks:

1. The condition ${a}_{n} \rightarrow 0$ is not sufficient: the series $\sum \frac{1}{n}$ is not convergent although $\frac{1}{n} \rightarrow 0$

2. In the case of a geometric series (such as the one given) if the absolute value of the ratio $| r | > 1$ the series diverges.