For what values of \(p\) will both series \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^{2p}}\) and \(\displaystyle \sum_{n=1}^\infty \left(\frac{p}{2}\right)^n\) converge?

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The first series is a \(p\)-series. These converge if the exponent is greater than 1.

$$ \sum_{n=1}^\infty \frac{1}{n^{2p}} $$ $$ 2p \gt 1 $$ $$ p \gt \frac{1}{2} $$

The second series is geometric and converges if the ratio is less than 1.

$$ \sum_{n=1}^\infty \left(\frac{p}{2}\right)^n $$ $$ \frac{p}{2} \lt 1 $$ $$ p \lt 2 $$