p-series
The series of 1 / n^p converges if p > 1 and diverges if p <= 1. With p = 1 it is the harmonic series, which diverges slowly.
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Analogy
A leaking tap: the water slows down fast enough (p > 1) or too slowly (p <= 1) to fill a finite bucket.
Example
Sum of 1/n^2: p = 2, converges (to about 1.645).Sum of 1/n: p = 1, diverges.Sum of 1/sqrt(n): p = 0.5, diverges.Sum of 1/n^3: converges.
In practice
A benchmark to compare other series against.
Practice
Flashcards
- For which p does the sum of 1/n^p converge?p > 1.
Questions people ask
Why does the harmonic series diverge if its terms go to zero?
The terms shrink too slowly. Grouping them shows each block of terms from 1/3 to 1/4, then 1/5 to 1/8, and so on adds more than 1/2. Endless blocks of at least 1/2 push the total past any limit, even though each single term becomes tiny.
How does the integral relate to p-series?
The integral test links the sum of 1 over n to the power p with the integral of 1 over x to the power p. We saw that integral converges exactly when p is above 1, so the series behaves the same way.