Question

What is the interval of convergence of `sum {n!( 8 x-7)^n}/{7^n}`?

Answer 1

The series is not convergent for any value of `x`.

Explanation:

Use the ratio test and evaluate:

`abs(a_(n+1)/a_n) = abs ( (( (n+1)! (8x-7)^(n+1))/7^(n+1))/(( n! (8x-7)^n)/7^n))`

`abs(a_(n+1)/a_n) = abs ( ((n+1)!)/(n!) 7^n/7^(n+1) (8x-7)^(n+1)/(8x-7)^n)`

`abs(a_(n+1)/a_n) = (n+1)/7 abs (8x-7)`

For any value of `x` we have therefore:

`lim_(n->oo) abs(a_(n+1)/a_n) = oo`

and the series is not convergent.