(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the Ratio Test for series to determine whether each of the following series converge or diverge. Make Reasoning Clear.

(a) [itex]\sum^{∞}_{n=1}\frac{3^{n}}{n^{n}}[/itex]

(b) [itex]\sum^{∞}_{n=1}\frac{n!}{n^{\frac{n}{2}}}[/itex]

2. Relevant equations

[itex]if\:lim_{n\rightarrow∞}(\frac{a_{n+1}}{a_{n}}) < 1 \Rightarrow\sum^{∞}_{n=1}\:-\:converges[/itex]

[itex]if\:lim_{n\rightarrow∞}(\frac{a_{n+1}}{a_{n}})\in[0,1)\Rightarrow \sum^{∞}_{n=1}\:-\:diverges[/itex]

[itex]0\leq \sum^{∞}_{n=1}(a_n)\leq \sum^{∞}_{n=1}(b_n)[/itex]

[itex]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\vdots[/itex]

[itex]if\:\sum^{∞}_{n=1}(a_n)\:-\:diverges\:\Rightarrow\:\sum^{∞}_{n=1}(b_n)\:-\:diverges[/itex]

[itex]if\:\sum^{∞}_{n=1}(b_n)\:-\:converges\:\Rightarrow\:\sum^{∞}_{n=1}(a_n)\:-\:converges[/itex]

3. The attempt at a solution

(a) I let [itex]a_{n}[/itex] = sequence of partial sums then plugged everything into ratio test formula.

I ended up with:

[itex]lim_{n\rightarrow∞}(\frac{a_{n+1}}{a_{n}})\:=\:3lim_{n\rightarrow∞}(\frac{n^{n}}{(n+1)(n+1)^{n}})[/itex]

I know that the limit equals 0 hence the series converges but not quite sure how to show that the limit cancels down to show 0 is "obvious"

Any help would be great, thanks.

(b) I let [itex]a_{n}[/itex] = sequence of partial sums then plugged everything into ratio test formula.

I ended up with:

[itex]lim_{n\rightarrow∞}(\frac{a_{n+1}}{a_{n}})\:=\:lim_{n\rightarrow∞}(\frac{n(\frac{n}{n+1})^{\frac{n}{2}}}{\sqrt{n+1}})\:+\:lim_{n\rightarrow∞}(\frac{(n)^{\frac{n}{2}}}{(n+1)^{\frac{n}{2}}\sqrt{n+1}})[/itex]

I know that:

[itex]lim_{n\rightarrow∞}(\frac{n(\frac{n}{n+1})^{\frac{n}{2}}}{\sqrt{n+1}})\:=\:∞[/itex]

And that:

[itex]lim_{n\rightarrow∞}(\frac{(n)^{\frac{n}{2}}}{(n+1)^{\frac{n}{2}}\sqrt{n+1}})\:=\:0[/itex]

Hence the overall limit = ∞ and so the series diverges; but, once again i'm not quite sure how to show that how the limits cancels down to show ∞ and 0 is "obvious"

Once again, any help would be great, thanks.

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# Homework Help: Using the Ratio Test to see if a series converges or diverges?

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