Given p>0,the series of#sum_ (n=1)^oo p^n*n^p# will diverge when the values of p is?

Given p>0,the series of#sum_ (n=1)^oo p^n*n^p# will diverge when the values of p is?

1 Answer
NickTheTurtle
Jun 25, 2018

#p<1# for positive #p#

Explanation:

Since all terms of the series is positive, the series will coverge if and only if it converges absolutely.

Denote #a_n=p^n n^p#. Then, by the ratio test, the series will converge absolutely if (assuming finite and positive #p#)

#lim_(n->oo)a_(n+1)/a_n<1#

#lim_(n->oo)(p^(n+1)(n+1)^p)/(p^n n^p)<1#

#lim_(n->oo)p(1+1/n)^p<1#

#p<1#

However, this is not complete. The ratio test is inconclusive if the limit equals #1#, at which #p=1#. Substitute this into the original sum, which gives

#sum_(i=1)^ooi#

which obviously diverges.

Thus, in order for the series to converge, #p<1# for positive #p#.

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