4.4 Q-3

Exercise 4.4 — Question 3

Harmonic Sequences and Series

A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence.

If $a_1,a_2,a_3,\dots$ is a harmonic sequence, then $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\dots$ is an arithmetic sequence.

General term of a harmonic sequence: $a_n=\frac{1}{a+(n-1)d}$ where $a=\frac{1}{a_1}$ is the first term of the corresponding AP and $d$ is the common difference of the corresponding AP.

Harmonic Mean (HM) between two numbers $a$ and $b$: $H=\frac{2ab}{a+b}$

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Arithmetic-Geometric Sequences

An arithmetic-geometric sequence is formed by multiplying corresponding terms of an arithmetic sequence and a geometric sequence.

If the arithmetic sequence has first term $a$ and common difference $d$, and the geometric sequence has first term $1$ and common ratio $r$, then the arithmetic-geometric sequence is: $a,(a+d)r,(a+2d)r^2,(a+3d)r^3,\dots$

General term: $T_n=[a+(n-1)d]r^{n-1}$

Sum to $n$ terms (using the subtraction method): $S_n=\frac{a-[a+(n-1)d]r^n}{1-r}+\frac{dr(1-r^{n-1})}{(1-r{)}^2},r\ne 1$

Sum to infinity (when $|r|<1$): $S_{\infty }=\frac{a}{1-r}+\frac{dr}{(1-r{)}^2}$

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Worked Example

Problem: Find the sum to infinity of the arithmetic-geometric series: $1+2\cdot \frac{1}{2}+3\cdot {\left(\frac{1}{2}\right)}^2+4\cdot {\left(\frac{1}{2}\right)}^3+\cdots$

Solution:

Here $a=1$, $d=1$, $r=\frac{1}{2}$, and $|r|<1$ so the series converges.

Using the formula: $S_{\infty }=\frac{a}{1-r}+\frac{dr}{(1-r{)}^2}$

$S_{\infty }=\frac{1}{1-\frac{1}{2}}+\frac{1\cdot \frac{1}{2}}{{\left(1-\frac{1}{2}\right)}^2}$

$S_{\infty }=\frac{1}{\frac{1}{2}}+\frac{\frac{1}{2}}{\frac{1}{4}}=2+2=4$