calculus II

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Given the series `sum_(n=1)^(oo) (1/(n+1)-1/(n+2))^n ` :

This series is absolutely convergent by the ratio test.

`sum a_n ` converges absolutely if ` ``lim_(n->oo) |a_(n+1)/a_n|<1 ` :

Rewrite the summand as ` ``(1/((n+1)(n+2)))^n ` ; then `a_(n+1)=(1/((n+2)(n+3)))^(n+1) ` .

`(1/((n+2)(n+3)))^(n+1)/(1/((n+1)(n+2)))^n `

`=((1/((n+2)(n+3)))^n*1/((n+2)(n+3)))/(1/((n+1)(n+2)))^n)`

`=((n+1)/(n+3))^n*1/((n+2)(n+3))`

`lim_(n->oo)((n+1)/(n+3))^n*1/((n+2)(n+3))<1 `

Approved by eNotes Editorial Team