How do you use the integral test to determine the convergence or divergence of $1+1/(2sqrt2)+1/(3sqrt3)+1/(4sqrt4)+1/(5sqrt5)+...$ ?

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Answer 1 · 2017-02-05T03:43:05

The series:

$sum_(n=1)^oo 1/(nsqrt(n))$

is convergent.

We have the series:

$sum_(n=1)^oo 1/(nsqrt(n))$

we can apply the integral test using as test function:

$f(x) = 1/(xsqrt(x)) = x^(-3/2)$

since the function is positive and decreasing in the interval $(1.+oo)$ and we have:

$lim_(x->oo) x^(-3/2) = 0$

Calculating the improper integral:

$int_1^oo x^(-3/2)dx = [-2x^(-1/2)]_1^oo$

$int_1^oo x^(-3/2)dx = 2-lim_(x->oo) -2/sqrt(x) = 2$

As the integral is convergent then the series is proven to be convergent.

How do you use the integral test to determine the convergence or divergence of 1+1/(2sqrt2)+1/(3sqrt3)+1/(4sqrt4)+1/(5sqrt5)+...1+1/(2sqrt2)+1/(3sqrt3)+1/(4sqrt4)+1/(5sqrt5)+...?