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)(adsbygoogle = window.adsbygoogle || []).push({});

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

conditions x≥0 and z≥1

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

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

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# Infinite series sum is positive

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