Is the following series 6-2+2/3-2/9+2/27-2/81 convergent or divergent?

I have this so far:
1/3(-1)^n or 1/3(-1)^(n-1).

I'm not sure which one it is.
Also, is a_i=6?

1 Answer
Apr 5, 2018

Read below.

Explanation:

We see that a given term, other than the first one, is the term before times negative one third.

The ratio is therefore -1/3.

A series is convergent if the absolute value of the ratio is less than 1.

abs(-1/3)=>1/3<1

This series is convergent.

Also, this series is in the form a+ar+ar^2+ar^3+....

This type of series is generally in the form:

a_n=a*r^(n-1) where r is the ratio and a is the first term.

Therefore, the formula for the nth term is:

a_n=6*(-1/3)^(n-1)

Also, the first term of the series is 6.

Therefore, a_1=6