How do you use the integral test to determine the convergence or divergence of #1+1/4+1/9+1/16+1/25+...#?
1 Answer
Our first step is to write the series in series notation. Notice that:
#1+1/4+1/9+1/16+1/25+...#
#=1/1^2+1/2^2+1/3^2+1/4^2+1/5^2+...#
#=sum_(n=1)^oo1/n^2#
So, we want to determine the convergence or divergence of the series
The integral test requires some conditions: for
#f(n)# has to be decreasing on#[N,oo)# #f(n)>=0# on the same interval#f(n)# is continuous on the interval
For
The series
#sum_(n=N)^oof(n)# converges if and only if the improper integral#int_N^oof(x)dx# converges to a finite number.
So, we want to see if
#int_1^oo1/x^2dx=lim_(brarroo)int_1^b1/x^2dx#
Then:
#=lim_(brarroo)int_1^bx^-2=lim_(brarroo) [x^-1/(-1)] _ 1^b=lim_(brarroo)[-1/x]_1^b#
Evaluating:
#=lim_(brarroo)(-1/b-(-1/1))=lim_(brarroo)(-1/b+1)#
As
#=0+1=1#
So,