# Infinite series sum is positive

1. Dec 8, 2011

### myname1234

I was trying to prove the following but couldn't succeed. Is there a systematic methods to prove that the following infinite sum is positive? (alternating series)

sum from n = 0 to ∞ of ((-1)$^{n}$* x$^{n+z}$) / (n+z)!

conditions x≥0 and z≥1

note: when x≤1, we can directly see that s$_{n}$- s$_{n+1}$ is positive for n ≥ 0. So the sum is positive.

However when x>1, s$_{n}$ = x$^{n+z}$ / (n+z)! monotonically increases first and then monotonically decreases to zero.

2. Dec 8, 2011

### McAfee

Conditions for the sum of alternating series are:
1. alternating
2. ingnoring signs the function is decreasing
3. the limit of the f(x) as x approaches infinity is 0.

Just by looking at the problem you don't use the alternating series test. You have to use the Ratio Test and find the interval of convergence.

3. Dec 8, 2011

### myname1234

sorry, but i am not looking whether the series is converging or not. I know it converges.

Also your condition number 2 is not necessary for an alternating series to converge.

I am interested in proving that the total sum is positive.