Question

Since lim n→ infinity (.............) = ...........

Answer 1

Notice that

`(4n+1)/(8n+2)=(4n+1)/(2(4n+1))=\frac12`

So by the root test,

`\displaystyle\lim_(n\to\infty)\sqrt[n]\left = \frac14 < 1`

and so the series converges (absolutely).