calculus - Convergence of series of positive terms: $sum_{n=2}^{infty}frac{1}{{(log n)}^{p}}$

I have applied Cauchy condensation test for to test the convergence the series $\sum_{n=2}^{\infty}\frac{1}{{(\log n)}^{p}}$, where p is constant, I got $\frac{1}{{\log 2}^{p}}\sum_{k=1}^{\infty}\frac{2^{k}}{k^{p}}$ . I do not understand for which value of p such that the original series is convergent. Also have used Cauchy integral test but did not solve the improper integral $\int_{2}^{\infty} {\frac{1}{(\log{x})^{p}}}dx$.
I do not understand the convergent or not, if it convergent what is the value of p will be. Please some one help me. Thanks