Convergence of P-Series/Real/Proof 1

Theorem

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

Then the $p$-series:

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

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

Proof

By the Cauchy Integral Test:

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

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

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.34 \ (3)$
• 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests