Use of ratio test and comparison test to determine converging series

Which of the following series converge, use ratio or comparison test to demonstrate:

1. $\sum_{n=1}^{\infty} \dfrac{n}{n^2+\cos^2(n)}$

2. $\sum_{n=1}^{\infty} \dfrac{(-10)^n}{4^{2n+1}(n+1)}$

3. $\sum_{n=1}^{\infty} \dfrac{n!}{3^n}$

4. $\sum_{n=1}^{\infty} \dfrac{n}{(n+3)^3}$

Help much appreciated! Thanks!

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Is this homework ? if so please add the homework tag to your question. what have you tried? where are you stuck ? –  Belgi Aug 26 '12 at 21:50
I have edited your post and set the sums in $\LaTeX$. There were ambiguities, so please have a look if it is what you intended. You can right-click on the sums to show the $\TeX$-Code, if you want to know how it is done. –  Gregor Bruns Aug 26 '12 at 21:52
Do you know the answers to any of these? What do you know about the tests you mention? –  marty cohen Aug 26 '12 at 22:05
@Belgi: No I'm going through past paper exam questions and these are a few I couldn't do. The hints were helpful and I looked through some textbooks so I've figured them all out! –  Enrico S Aug 28 '12 at 20:12
@GregorBruns: Thank you so much! Yes all correct, I just started using this website so I had no clue how to do that. –  Enrico S Aug 28 '12 at 20:15

A slightly different set of HINTS:

1. $n^2+\cos^2n\le2n^2$, so how is $\dfrac{n}{n^2+\cos^2n}$ related to $\dfrac{n}{2n^2}$?

2. The ratio test often works well when you have products or quotients of exponentials with the index of summation in the exponent.

3. As mixedmath said, the ratio test also behaves well with factorials of linear functions of the index of summation.

4. $n<n+3$, so how is $\dfrac{n}{(n+3)^3}$ related to $\dfrac{n}{n^3}$? Alternatively, $$\sum_{n=1}^\infty\frac{n}{(n+3)^3}=\sum_{n=4}^\infty\frac{n-3}{n^3}<\dots\;?$$

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Thank you for the hints they were very helpful! I'm still unsure about one thing: How do you identify whether to use the ratio or comparison test just from looking at the series? (aside from factorials and quotients of exponentials like you mentioned) Is the best way just to try both and see which works out? –  Enrico S Aug 29 '12 at 16:41
@Enrico: Mostly through experience, I think. For instance, I see $\dfrac{n}{n^2+\cos^2n}$ and immediately think that it’s between $\dfrac{n}{n^2}=\dfrac1n$ and $\dfrac{n}{n^2+1}\approx\dfrac1n$, so I expect to be able to compare with the harmonic series. I see $\dfrac{n}{(n+3)^3}$ and immediately think that it’s a lot like $\dfrac{n}{n^3}=\dfrac1{n^2}$ and expect (maybe after a bit of algebraic juggling) to be able to compare it usefully with $\sum\frac1{n^2}$. –  Brian M. Scott Aug 29 '12 at 16:51

HINTS

1. Note that $\cos^2(x)$ is always between $0$ and $1$. That sounds like a good thing to compare against.
2. Just do it. There's a negative, so you might use the "alternating series test." It looks sort of geometric, so you might use the ratio test. Of the basic divergence test.
3. A lesson for the future - factorials are hard to play with in limits and in series. The ratio test works nicely with factorials, thought (why?) so in general, you might want to associate the ratio test with factorials.
4. Just do it. There's no need to do anything slick. Or alternately note that $n+3 > n$.
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