nLab harmonic series

This entry is about a notion in mathematics. For the related notion i physics and music, see at overtone series.

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Definition

Let $f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ defined as

$$f(n)(x) \coloneqq \sum_{i = 1}^n \frac{\cos(n x)}{n}$$

be a Fourier series. The harmonic series is defined as

$$h(n) \coloneqq f(n)(2 \pi m) = \sum_{i = 1}^n \frac{\cos(2 \pi m n)}{n} = \sum_{i = 1}^n \frac{1}{n}$$

for any integer $m \in \mathbb{Z}$ and the alternating harmonic series is defined as

$$h_\mathrm{alt}(n) \coloneqq - f(n)(2 \pi m + \pi) = - \sum_{i = 1}^n \frac{\cos(2 \pi m n + \pi n)}{n} = - \sum_{i = 1}^n \frac{(-1)^{n}}{n} = \sum_{i = 1}^n \frac{(-1)^{n + 1}}{n}$$

for any integer $m \in \mathbb{Z}$.

The harmonic series diverges:

$$\lim_{n \to \infty} h(n) = \infty$$

while the alternating harmonic series converges:

$$\lim_{n \to \infty} h_\mathrm{alt}(n) = \ln(2)$$

where $\ln$ is the natural logarithm. This is because the Fourier series above converges everywhere in $\mathbb{R}$ that is apart from an integer multiple of $2 \pi$.

See also

• series

• convergence

• Fourier analysis

References

• Wikipedia, Harmonic series (mathematics)