Harmonic sequence

The first 10 members of the harmonic sequence

The harmonic sequence is the mathematical sequence of numbers of the reciprocal values ​​of the positive whole numbers, i.e. the sequence

${\displaystyle 1,\;{\frac {1}{2}},\;{\frac {1}{3}},\;{\frac {1}{4}},\;{\frac {1}{5}},\cdots }$

with the general limb

${\displaystyle a_{n}={\frac {1}{n}}\quad n\geq 1}$.

Each member of the harmonic sequence with is the harmonic mean of its neighboring members. The summation of the following terms results in the harmonic series . ${\displaystyle n\geq 2}$

The alternating harmonic sequence has the general term

${\displaystyle a_{n}={\frac {\left(-1\right)^{(n+1)}}{n}}\quad n\geq 1}$.

For is the generalized harmonic sequence ${\displaystyle k\in \mathbb {N} }$

${\displaystyle \left(a_{n}\right)_{n\in \mathbb {N} }=\left({\frac {1}{n^{k}}}\right)_{n\in \mathbb {N} }=\left(1,\,{\tfrac {1}{2^{k}}},\,{\tfrac {1}{3^{k}}},\,{\tfrac {1}{4^{k}}},\,{\tfrac {1}{5^{k}}},\,\ldots \right)}$

properties

• The harmonic sequence converges to zero: . ${\displaystyle \lim _{n\to \infty }{\tfrac {1}{n}}=0}$

• The harmonic sequence is monotonically falling and has only strictly positive sequence terms.
• The maximum of the terms of the sequence and thus the supremum is 1. The infimum of the terms of the sequence is 0, which is not assumed by the sequence.

swell

1. ↑ University of Heidelberg : Sequences and Series Series (F3) - accessed on January 3, 2015.
2. ↑ University of Heidelberg: Sequences and Series Series (F7) - accessed on January 3, 2015.