Question

.converge or diverge?  If it converges, to what value does it converge?

Answer 1

Given the series;

`\sum ^(\infty)_(n\mathop=0)3((1)/(5))^(n-1)`

To obtain the sum of the series above and decide if it converges or diverges, we will

`\begin{gathered} \sum ^(\infty)_{n\mathop{=}0}3((1)/(5))^(n-1)=\sum ^(\infty)_{n\mathop{=}0}3(5)^(-(n-1)) \\ =\sum ^(\infty)_{n\mathop{=}0}3(5)^((1-n)) \\ =\sum ^(\infty)_{n\mathop{=}0}15*5^(-n) \\ =15\sum ^(\infty)_{n\mathop{=}0}((1)/(5))^n \end{gathered}`

Simplify the resulting geometric series and decide if it converge or diverge

`\sum ^(\infty)_{n\mathop{=}0}((1)/(5))^n\Rightarrow is\text{ an infinite geometric series, with first term a= 1 and common ratio r=}(1)/(5)`

Solve for the sum to infinity of the geometric series

`S_(\infty)=(a)/(1-r)=(1)/(1-(1)/(5))=(1)/((4)/(5))=(5)/(4)`

The sum of the series wil be

`15\sum ^(\infty)_{n\mathop{=}0}((1)/(5))^n\Rightarrow15*(5)/(4)=(75)/(4)`

Hence,

`\begin{gathered} \sum ^(\infty)_{n\mathop{=}0}3((1)/(5))^(n-1)=(75)/(4) \\ \text{The series converges} \end{gathered}`