Question

Find the values of x for which the series converges. (Enter your answer using interval notation.) [infinity] (x + 7)n n = 1

Answer 1

x belongs to (-8,-6)

Explanation:

Given is a series in x as

`\Sigma _1^(\infty) (x+7)^n\\=(x+7)+(x+7)^2+(x+7)^3+(x+7)^4+(x+7)^5+...(x+7)^n+...`

we find that first term is x+7 and each successive term is multiplied by x+7

In other words this is a geometric series with common ratio as

`r=x+7\\a=x+7`

An infinite geometric series converges only for

|r|<1

Hence here we have if series converges,

`|x+7|<1\\-1<x+7<1\\-8<x<-6`

For all values of x lying in the open interval (-8,-6) the series converges.