Harmonic Series

Definition

Convergence

Alternative Proofs of Harmonic Series Divergence

Method 1: Grouping Method

Group the harmonic series as follows:

$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \cdots$

$= 1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \cdots$

$> 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots$

$= 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots$

$= 1 + \frac{1}{2} \times \infty = +\infty$

Therefore, the harmonic series diverges.

Method 2: Comparison Method

Since $\frac{1}{n} \geq \frac{1}{n+1}$, we have:

$\sum_{n=1}^{\infty} \frac{1}{n} \geq \sum_{n=1}^{\infty} \frac{1}{n+1} = \sum_{n=2}^{\infty} \frac{1}{n}$

Now, $\sum_{n=2}^{\infty} \frac{1}{n}$ differs from $\sum_{n=1}^{\infty} \frac{1}{n}$ by only a constant term, so if $\sum_{n=1}^{\infty} \frac{1}{n}$ converges, then $\sum_{n=2}^{\infty} \frac{1}{n}$ also converges, which contradicts the divergence of the harmonic series.

Exercises

Exercise 1

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n}$.

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Summary

Symbols Used in This Article