Difference between revisions of "Harmonic series"
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− | There are several types of '''harmonic series'''. | + | Generally, a '''harmonic series''' is a [[series]] whose terms involve the [[reciprocal]]s of the [[positive integer]]s. |
+ | |||
+ | There are several sub-types of '''harmonic series'''. | ||
The the most basic harmonic series is the infinite sum | The the most basic harmonic series is the infinite sum | ||
− | <math> | + | <math>\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots</math> |
This sum slowly approaches infinity. | This sum slowly approaches infinity. | ||
The alternating harmonic series, | The alternating harmonic series, | ||
− | <math> | + | <math>\sum_{i=1}^{\infty}\frac{(-1)^{i+1}}{i}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots</math> , though, approaches <math>\ln 2</math>. |
− | |||
− | |||
The [[zeta-function]] is a harmonic series when the input is one. | The [[zeta-function]] is a harmonic series when the input is one. | ||
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== How to solve == | == How to solve == | ||
− | ''' | + | === Harmonic Series === |
+ | |||
+ | It can be shown that the harmonic series diverges by grouping the terms. We know that the first term, <math>1</math>, added to the second term, <math>\frac{1}{2}</math> is greater than <math>\frac{1}{2}</math>. We also know that the third and and fourth terms, <math>\frac{1}{3}</math> and <math>\frac{1}{4}</math>, add up to something greater than <math>\frac{1}{2}</math>. And we continue grouping the terms between powers of two. So we have | ||
+ | <math>\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots=\left(1+\frac{1}{2}\right)+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdots \ge \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots \to \infty</math> | ||
+ | |||
+ | === Alternating Harmonic Series === | ||
+ | |||
+ | === General Harmonic Series === | ||
+ | <math>\sum_{i=1}^{\infty}\frac{1}{ai +b}</math> is the general harmonic series, where each term is the reciprocal of a term in an arithmetic series. | ||
+ | |||
+ | '''Case 1:''' <math>a\ge b</math> | ||
+ | |||
+ | <math>ai+a\ge ai+b</math> | ||
+ | |||
+ | <math>\frac{1}{ai+b}\ge\frac{1}{ai+a}=\frac{1}{a}\left(\frac{1}{i+1}\right)</math> | ||
+ | |||
+ | <math>\sum_{i=1}^{\infty}\frac{1}{ai+b}\ge\frac{1}{a} \left(\sum_{i=1}^{\infty}\frac{1}{i+1}\right)\to\infty</math> | ||
+ | |||
+ | '''Case 2:''' <math>a<b</math> | ||
+ | |||
+ | <math>ai+b<bi+b</math> | ||
+ | |||
+ | <math>\frac{1}{ai+b}>\frac{1}{bi+b}=\frac{1}{b}\left(\frac{1}{i+1}\right)</math> | ||
− | + | <math>\sum_{i=1}^{\infty}\frac{1}{ai+b}\ge\frac{1}{b} \left(\sum_{i=1}^{\infty}\frac{1}{i+1}\right)\to\infty</math> | |
− | |||
− | + | Thus, <math>\sum_{i=1}^{\infty}\frac{1}{ai+b}=\infty</math> |
Latest revision as of 20:52, 13 March 2022
Generally, a harmonic series is a series whose terms involve the reciprocals of the positive integers.
There are several sub-types of harmonic series.
The the most basic harmonic series is the infinite sum
This sum slowly approaches infinity.
The alternating harmonic series,
, though, approaches
.
The zeta-function is a harmonic series when the input is one.
Contents
How to solve
Harmonic Series
It can be shown that the harmonic series diverges by grouping the terms. We know that the first term, , added to the second term,
is greater than
. We also know that the third and and fourth terms,
and
, add up to something greater than
. And we continue grouping the terms between powers of two. So we have
Alternating Harmonic Series
General Harmonic Series
is the general harmonic series, where each term is the reciprocal of a term in an arithmetic series.
Case 1:
Case 2:
Thus,