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

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Answer 1 · 2018-04-15T17:47:55

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

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.