Question

I need help solving/answering this, it’s from my ACT prep guide

Answer 1

Given the sum:

`\sum ^(\infty)_(n\mathop=1)(3n^5)/(6n^6+1)`

Analyzing the argument:

`(3n^5)/(6n^6+1)=(3n^5)/(n^6(6+(1)/(n^6)))=(1)/(n)\cdot((3)/(6+(1)/(n^6)))`

Where:

`\lim _(n\to\infty)((3)/(6+(1)/(n^6)))=(3)/(6+0)=(1)/(2)\text{ (Bounded)}`

It is known that the harmonic sum diverges:

`\sum ^(\infty)_(n\mathop=1)(1)/(n)=\infty`

And we have a multiplication between a term that diverges and a bounded term, so we can conclude that the product diverges. Then:

`\begin{gathered} \sum ^(\infty)_(n\mathop=0)(1)/(n)\cdot((3)/(6+(1)/(n^6)))=\infty \\ \\ \Rightarrow\sum ^(\infty)_{n\mathop{=}1}(3n^5)/(6n^6+1)=\infty \end{gathered}`