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

1 Answer
Oct 27, 2016

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.