Convergence of P-Series/Real/Proof 1

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Theorem

Let $p \in \R$ be a real number.

Then the $p$-series:

$\displaystyle \sum_{n \mathop = 1}^\infty n^{-p}$

is convergent if and only if $p > 1$.


Proof

By the Integral Test:

$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^x}$ converges if and only if the improper integral $\displaystyle \int_1^\infty \frac {\d t} {t^x}$ exists.

The result follows from Integral to Infinity of Reciprocal of Power of x.

$\blacksquare$


Sources