How do you use the integral test to determine the convergence or divergence of $sum_(n=1)^(infty) 3/(n^(5/3))$ ?

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Answer 1 · 2018-02-04T10:25:38

The series converges

The series can be written as

$sum_(n=1)^oo3/(n^(5/3))=3sum_(n=1)^oo1/(n^(5/3))$

Let $f(x)=1/(x^(5/3$

This function $f(x)$ is continuous, positive and decreasing on the Interval $(1,+oo)$ , so we can apply the integral test

The integral

$int_1^(+oo)f(x)dx=int_1^(+oo)(1dx)/(x^(5/3)$ converges as

$lim_(x->+oo)int_1^x(x^(-5/3)dx)=lim_(x->+oo)[-3/2x^(-2/3)]_1^x$

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

$=0+3/2$

$=3/2$

Therefore,

$sum_(n=1)^oo3/(n^(5/3))$ converges to $=3*3/2=9/2$

How do you use the integral test to determine the convergence or divergence of sum_(n=1)^(infty) 3/(n^(5/3))sum_(n=1)^(infty) 3/(n^(5/3))?