Home > Calculus > Integral Test for Convergence of an Infinite Series

How do you use the integral test to determine the convergence or divergence of #1+1/sqrt2+1/sqrt3+1/sqrt4+...#?

Answer 1

Christopher Allers

The series is divergent and therefore has no finite sum.

This is defined by the formula

#t_n = 1/sqrt(n)#

Therefore, if the integral

#int_1^(oo) 1/sqrt(n) dn#

Has a calculable value, then the series converges. The integral can be rewritten as

#S = lim_(t->oo) int_1^t n^(-1/2) dn#

#S = [2n^(1/2)]_1^t#

#S = lim_(t->oo) 2sqrt(t) - lim_(t->oo) 2(1)^(1/2)#

The first limit obviously has value of #oo#, therefore the series diverges and has no finite sum.

Hopefully this helps!

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.