The Harmonic Series
Table of Contents

The Harmonic Series

One very famous series known as the harmonic series denoted $\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + ...$ is a classic example of a divergent series surprisingly enough. At first we might suspect that this series is convergent since we know the series $\sum_{n=1}^{\infty} \frac{1}{n^2} = 2$ is convergent. In fact, it is not and we will show this.

Consider the indefinite integral $\int_{1}^{\infty} \frac{1}{x} \: dx$. When we evaluate this indefinite integral we get that:

\begin{align} \int_{1}^{\infty} \frac{1}{x} \: dx \\ = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x} \: dx \\ = \lim_{b \to \infty} \left ( \ln x \right )_{1}^{b} \\ = \lim_{b \to \infty} \left ( \ln b - \ln 1 \right ) \\ = \lim_{b \to \infty} \ln b = \infty \end{align}

Now consider the harmonic series as illustrated in the following diagram. We note the area of each of the rectangles corresponds to the area of a term in the sum. For example, $1$ corresponds to the area $1 \cdot 1$. $\frac{1}{2}$ corresponds to the area $1 \cdot \frac{1}{2}$, etc…


As we can see, the sum of the pink rectangles is greater than the area under the curve $f(x) = \frac{1}{x}$. We know that the area under the curve $\frac{1}{x}$ is $\infty$ though, so we deduce that $\sum_{n=1}^{\infty} \frac{1}{n} = \infty$.

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