Question

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?

Answer 1

`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`.