P-Series Convergence Condition

Question

For what values of $$p$$ does the p-series $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ converge?

| $$p$$ Value | Behavior of the Series |
|:---|:---|
| 0.5 | Diverges |
| 1 | Diverges (Harmonic) |
| 2 | Converges |

Accepted Answer

Converges for $$p > 1$$

Answer

Always converges regardless of the value of $$p$$, since the terms decrease.

Answer

Converges for $$p < 1$$

Answer

Converges for $$p \ge 1$$

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