The P-series | The Infinite Series Module - UBC Blogs

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The Infinite Series Module A Set of Open Resources on Infinite Series Skip to content
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    • Unit 2: Convergence Tests
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      • The Alternating Series Test
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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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    • Home
    • About the ISM
    • Units
      • Unit 1: Sequences and Series
      • Unit 2: Convergence Tests
        • The Divergence Test
        • The Integral Test
          • A Motivating Problem for The Integral Test
          • A Second Motivating Problem for The Integral Test
          • Theorem: The Integral Test
          • An Integral Test Flowchart
          • Integral Test Example
          • Example: Integral Test with a Logarithm
          • The p-series
          • Videos on the Integral Test
          • Final Thoughts on The Integral Test
        • The Alternating Series Test
        • The Ratio Test
        • Review of Convergence Tests
      • Unit 3: Power Series
    • Appendices
    • Download
    • Quizzes
    • Contact
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