Infinite series sum is positive

[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.

 

[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.

 

[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.