I discovered the following sequence, ∑ k = 1 ∞ exp ⁡ ( − p k ) k 1 − p where p ≥ 0

Question

I discovered the following sequence,      ∑          k      =      1        ∞              exp      ⁡      (      −      p      k      )              k              1        −        p            where   p  ≥  0

Boatein · Accepted Answer

The integral test is useful here and note that      ∫    1    ∞        x          p      −      1            e          −      p      x           d  x  ≤      ∫    0    ∞        x          p      −      1            e          −      p      x           d  xwhich is the Gamma distribution and finite for all   p  &amp;gt;  0. Hence it converges for all   p  &amp;gt;  0. The case   p  =  0 is clearly divergent by the p-test.

I discovered the following sequence, <munderover> ∑ <mrow class="MJX-Te

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Answer & Explanation

New Questions in Calculus 2