The series ∑n=1∞nkrn coverges when r∈(0,1) and diverges when r&gt;1. This is true regardless of the value of the constant k. When r=1 the series is a p-series. It converges if k&lt;−1 and diverges otherwise Each of the series below can be compared to a series of the form ∑n=1∞nkrn. For each series determine the best value of r and decide whether the series converges.I am stuck on this question ∑n=1∞(3n2+4n+2−2n7n+2+4n+5n)2I tried using 2−2n7n+2 and ended up with a value of r of 14⋅7 but this didn't work.

Question

The series ∑n=1∞nkrn coverges when r∈(0,1) and diverges when r&amp;gt;1. This is true regardless of the value of the constant k. When r=1 the series is a p-series. It converges if k&amp;lt;−1 and diverges otherwise Each of the series below can be compared to a series of the form ∑n=1∞nkrn. For each series determine the best value of r and decide whether the series converges.I am stuck on this question ∑n=1∞(3n2+4n+2−2n7n+2+4n+5n)2I tried using 2−2n7n+2 and ended up with a value of r of 14⋅7 but this didn&#039;t work.

soanooooo40 · Accepted Answer

Hint:  4n2+4n+2−2n∼3n2(n→+∞)  7n+2+4n+5n∼7n+2(n→+∞)  the general term of your series is equivalent to  9n472n+4  and has the same nature as  ∑n4(149)n

limacarp4 · Accepted Answer

Note that&amp;nbsp;limn→∞(3n2+4n+2−2n7n+2+4n+5n)2n47−2n=limn→∞(7n(3n2+4n+2−2n)n2(7n+2+4n+5n))2&amp;nbsp;=limn→∞(3+4n−1+2−2nn−272+4n7−n+5n7−n)2&amp;nbsp;=3274&amp;nbsp;So, since the series&amp;nbsp;∑n=1∞n472n&amp;nbsp;converges, your series converges too.

The series \sum_{n=1}^\infty n^kr^n coverges when r

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