Question

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`?

Answer 1

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 `n`th term is:

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

Also, the first term of the series is `6`.

Therefore, `a_1=6`