The p-series

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Problem

For what values of p does the infinite series

converge?

Complete Solution

Step (1): Consider p > 0 and p ≠ 1

When p > 0 and p ≠ 1, the function

is continuous, decreasing, and positive when x is in the interval [1,∞). Using the integral test,

Therefore, the infinite series converges when p > 1, and diverges when p is in the interval (0,1).

Step (2): Consider p ≤ 0 and p = 1

If p=1, then we have the harmonic series

which we know diverges.

If p ≤ 0, the infinite series diverges (by the divergence test).

Therefore, the given series only converges for p > 1.

The p-Series

The result of this example can be summarized as follows.

The p-Series

The p-series

is convergent if p > 1 and divergent if p ≤ 1.

Much like a geometric series, we can use this result to determine whether a given infinite series converges by inspection. For example, the infinite series

diverges because it is a p-series with p equal to 1/2 (you may want to let u=(1+k) to see this).


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Next: Videos on the Integral Test



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