WOLFRAM|DEMONSTRATIONS PROJECT

The P-Series Theorem

p

2

The area under the graph of

p

x

is the integral

∞

∫

1

p

x

x

, which is infinite when

p≤1

and finite when

p>1

. The integral test implies that the

p

-series

∞

∑

n=1

1

p

n

diverges when

p≤1

and converges when

p>1

.