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.

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