Question

Does the following infinite series converge or diverge? Explain your answer. 1/5 + 1/15 + 1/45 + 1/81...

A.It diverges; it has a sum.


B.It converges; it does not have a sum


C.It diverges; it does not have a sum.


D.It converges; it has a sum.

Answer 1

It converges; it has a sum.

Answer 2

Answer with explanation:

The given series is

`(1)/(5)+(1)/(15)+(1)/(45)+(1)/(81)+.....`

The given series is a geometric sequence,whose common ratio is equal to

`R=\frac{2^(nd)\text{term}}{1^st \text{term}}\\\\R=((1)/(15))/((1)/(5))\\\\R=(5)/(15)\\\\R=(1)/(3)`

Sum to Infinity is given by the formula

`S_(\infty)=\frac{\text{First term}}{1-\text{Common ratio}} \text{or}\frac{\text{First term}}{\text{Common ratio}-1} \\\\S_(\infty)=((1)/(5))/(1-(1)/(3))\\\\S_(\infty)=((1)/(5))/((2)/(3))\\\\S_(\infty)=(3)/(10)`

As the sum of series is Finite, that is having a single value, so the series is Convergent.

If it has more than one sum, it would have been Divergent.

Option D: It converges; it has a sum.