Question

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

Answer 1

For a series to have a finite sum the general term of the series must tend to zero when `n rarr oo`

Explanation:

In other words, `a_n rarr 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 (7/3)^n`, which does not tend to zero when `n` tends to `oo`. 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 rarr 0` is not sufficient: the series `sum 1/n` is not convergent although `1/n rarr 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.