1. Comparison Test Problems

Hello All: I am discovering I am really bad at these comparison test problems -- of the 15 homework assignments, I have 4/5 that I can't do: please help me! While I don't want to beg for the answers, it would be v. helpful if someone could give me advice in setting these problems up/tips on completing them:

#20: Sum (n=1 to Infinity) of (n + 4^n)/(n + 6^n)

I get really confused in the whole "an/bn" business!

#26: Sum (n=1 to Infinity) of n + 5/((n^7 + n^2)^(1/3))

#28: Sum (n=1 to Infinity) of (e^(1/n))/n

#34: Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.

Sum (n=1 to Infinity) of (sin^2(n))/(n^3)

I said that (sin^2(n))/(n^3) is less than or equal to 1/(n^3), Rn is less than or equal to Tn is less than or equal to 1/2n^2, n = 10, R10 is less than or equal to .005, is this correct? How do I approximate the sum of the series?

Eternal gratitude to whomever can assist with these problems! I really have been struggling with the comparison test -- everything else this chapter on series hasn't been too hard. Any advice is desperately and gratefully appreciated!!!!

2. Originally Posted by Sprintz
Hello All: I am discovering I am really bad at these comparison test problems -- of the 15 homework assignments, I have 4/5 that I can't do: please help me! While I don't want to beg for the answers, it would be v. helpful if someone could give me advice in setting these problems up/tips on completing them:

#20: Sum (n=1 to Infinity) of (n + 4^n)/(n + 6^n)

I get really confused in the whole "an/bn" business!
$\displaystyle \frac{n+4^n}{n+6^n}<\frac{n+4^n}{n+6^n}$

and for large $\displaystyle n$: $\displaystyle n<4^n$, so:

$\displaystyle \frac{n+4^n}{n+6^n}<\frac{n+4^n}{6^n}<\frac{2\time s 4^n}{6^n}=2\left(\frac{1}{3}\right)^n$

CB

3. Originally Posted by Sprintz
Hello All: I am discovering I am really bad at these comparison test problems -- of the 15 homework assignments, I have 4/5 that I can't do: please help me! While I don't want to beg for the answers, it would be v. helpful if someone could give me advice in setting these problems up/tips on completing them:

[snip]

#28: Sum (n=1 to Infinity) of (e^(1/n))/n

[snip]
Substitute the series expansion $\displaystyle e^{1/n} = 1 + \frac{1}{n} + \frac{1}{2 n^2} + ....$.

Then you get $\displaystyle \sum_{n=1}^{+\infty} \frac{1}{n} + \sum_{n=1}^{+\infty} \frac{1}{n^2} + \sum_{n=1}^{+\infty} \frac{1}{2n^3} + ....$

Now look at the first sum.

Edit: Oops, I just noticed the title of your post. OK, note that $\displaystyle \frac{e^{1/n}}{n} > \frac{1}{n}$ ....