Convergence of 2 + and 1 - alternating series

[benf.stokes]

Homework Statement

The problem asks you if the series:
1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}...

converges or diverges

Homework Equations

The Attempt at a Solution

I tried to apply the Leibniz rule but I realized it can't be applied. Is there a transformation of this series that enables the use of Leibniz rule?

 

[Dick]

If you write f(n)=1/n-1/(n+1)+1/(n+2), then the series becomes 1+f(2)+f(5)+f(8)+... Can you say something about the behavior of f(n) that might make you think it converges or diverges?

 

[benf.stokes]

lim n tend to infinity of f(n) is different from zero and thus the series does not converge?

 

[Dick]

lim n tend to infinity of f(n) is different from zero and thus the series does not converge?

Why do you think lim n->infinity f(n) isn't zero??

 

[benf.stokes]

Sorry. I misread your post. It is indeed zero. How then is this rewriting useful? f(n) is always positive and the ratio test is useless as it gives r=1

 

[Dick]

Sorry. I misread your post. It is indeed zero. How then is this rewriting useful? f(n) is always positive and the ratio test is useless as it gives r=1

Try a comparison test. Can you make a simple estimate of the size of f(n) for large n? Combine the terms in f(n).

 

[benf.stokes]

f(n)=\frac{n^2+2*n+6}{n*(n+1)*(n+2)}=\frac{n+1}{n*(n+2)}+\frac{5}{n*(n+1)*(n+2)} > \frac{n+1}{n*(n+2)}>\frac{1}{n}

and thus by the comparison test the series diverges?

 

[Dick]

f(n)=\frac{n^2+2*n+6}{n*(n+1)*(n+2)}=\frac{n+1}{n*(n+2)}+\frac{5}{n*(n+1)*(n+2)} > \frac{n+1}{n*(n+2)}>\frac{1}{n}

and thus by the comparison test the series diverges?

I think your comparisons might be a bit sloppy. But you've got an n^2 in the numerator and an n^3 in the denominator, so you can certainly show f(n)>C/n for some value of C. You also have a be a little careful because the series doesn't sum over all n. Only over n=2,5,8,etc. So you'll need an argument of show that series diverges as well. But sure, once you have f(n)>C/n, the series diverges.

 

[benf.stokes]

Thanks for everything . One last thing though: how should I do the comparison so that it's less sloppy?