How do you use the integral test to determine if #1/2+1/5+1/10+1/17+1/26+...# is convergent or divergent?

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Answer 1 · 2017-01-11T02:46:09

The series: #sum_(n=1)^oo 1/(n^2+1)# is convergent.

The series is:

#sum_(n=1)^oo 1/(n^2+1)#

so we can use for the integral test the function:

#f(x) = 1/(1+x^2)#

which is positive, decreasing and infinitesimal for #x->oo#, thus satisfying all the hypotheses of the integral test.

Calculating the integral:

#int_1^oo 1/(1+x^2)dx = [arctan x]_1^oo = pi/2-pi/4 =pi/4#

So the series is convergent.