Given these two series:
[tex]\begin{gathered} \sum ^{\infty}_{n\mathop=1}\frac{1}{n^{2p}} \\ \sum ^{\infty}_{n\mathop{=}1}(\frac{p}{2})^n \end{gathered}[/tex]
The second one is a way to write the geometric series, where r = p/2:
[tex]\text{Geometric series}\colon\sum ^{\infty}_{n\mathop=1}r^n[/tex]
This series converges only if 0 < r < 1, so the condition of convergence of the second series is:
[tex]\begin{gathered} 0\leq\frac{p}{2}<1 \\ 0\le p<2\ldots(1) \end{gathered}[/tex]
Now, for the first one, we know that the series diverges when 2p = 1, leading to the so-called Harmonic Series:
[tex]\sum ^{\infty}_{n\mathop=1}\frac{1}{n}=\infty[/tex]
So, the condition of convergence should be:
[tex]\begin{gathered} 2p>1 \\ p>\frac{1}{2}\ldots(2) \end{gathered}[/tex]
Combining these two conditions, (1) and (2), leads to:
[tex]\frac{1}{2}
