Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.sum_{n=1}^inftyfrac{1}{(2n+3)^3}

Cabiolab 2020-10-18 Answered

Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.
n=11(2n+3)3

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Expert Answer

tabuordy
Answered 2020-10-19 Author has 91 answers
Integral test:
Let f(x) is positive decreasing continuous function on [k,) and f(n)=an then
If kf(x)dx is convergent then n=kan is also convergent.
If kf(x)dx is divergent then n=kan is also divergent.
So we can use integral test in given function.
Given that
n=11(2n+3)3
Integral test:
Here f(x)=1(2n+3)3 so
11(2n+3)3dx=1(2x+3)3dx=12[(2x+3)22]1
=14[1(2x+3)2]1
=14[1250]
=1100
Which is finite, so 11(2x+3)3dx is convergent
Since 11(2x+3)3dx is convergent so using integral test n=11(2n+3)3 si also convergent.
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