Alternating series

1. Nov 25, 2013

Jbreezy

1. The problem statement, all variables and given/known data
Ʃ (-1)^n [ n+ln(n) / n-ln(n)] from n = 2 to infinity.

2. Relevant equations

I looked at the limit first because I thought lnn was very slow function. n would go faster.

3. The attempt at a solution

limit n --> ∞ [ n+ln(n) / n-ln(n)] = 1 so it diverges.
Limit is not 0 so it violates the one of the conditions? OK ? OR wrong?
Thanks,

2. Nov 25, 2013

Dick

Yes, it doesn't converge. If $|x_n|$ doesn't converge to 0, then $\Sigma x_n$ doesn't converge. That's true whether the series is alternating or not. You still have to prove its limit isn't zero. Just saying 'slow function' doesn't do the job.

3. Nov 25, 2013

Ray Vickson

USE PARENTHESES! What you have written is
$$\lim_{n \to \infty} n + \frac{\ln(n)}{n} - \ln(n) = 1 \, \leftarrow \text{ false}$$
Perhaps you meant
$$\lim_{n \to \infty} \frac{n + \ln(n)}{n - \ln(n)} = 1,$$
which is true. What would be so hard about writing (ln(n)+n}/(ln(n)-n), or [ln(n)+n]/[ln(n)-n], if that is what you really meant?