42. Infinite Series

Dated: 30-10-2024

Definition

An infinite series is an expansion that can be written in the form

\[\sum_{i = 1}^{\infty} u_i = u_1 + u_2 + \ldots\]

The \(u_1\) and \(u_2\) etc are called the terms of the series.

Example

Imagine we have \(0.333\ldots\)
Then we can re-write it as follows

\[0.3 + 0.03 + 0.003 + \ldots\]

\[\frac{3}{10} + \frac{3}{10^2} + \frac{3}{10^3} + \ldots\]

We do know that \(\frac 1 3 = 0.333\ldots\)
Therefore, this series should converge to \(\frac 1 3\).

\[s_1 = \frac{3}{10} = 0.3\]

\[s_2 = \frac{3}{10} + \frac{3}{10^2} = 0.33\]

\[s_3 = \frac{3}{10} + \frac{3}{10^2} + \frac{3}{10^3} = 0.333\]

This means that as we keep adding more and more terms, our approximation gets more and more close to the actual answer.
Therefore, we would just need sum¹ of some finite \(n\) values and our approximation should be good enough.

\[S_n = \frac{3}{10} + \frac{3}{10^2} + \frac{3}{10^3} + \cdots + \frac{3}{10^n}\]

Applying limit,² we get

\[\lim_{n\rightarrow+\infty}S_{n}=\lim_{n\rightarrow+\infty}\left(\frac{3}{10}+\frac{3}{10^{2}}+\frac{3}{10^{3}}+\cdots+\frac{3}{10^{n}}\right)\]

Dividing by \(10\), we get

\[\frac{1}{10}S_n = \frac{3}{10^2} + \frac{3}{10^3} + \cdots + \frac{3}{10^n} + \frac{3}{10^{n+1}}\]

Subtracting the original from this new series, we get

\[S_n - \frac{1}{10}S_n = \frac{3}{10} - \frac{3}{10^{n+1}}\]

\[\frac{9}{10}S_n = \frac{3}{10}\left(1 - \frac{1}{10^n}\right)\]

\[S_n = \frac{1}{3}\left(1 - \frac{1}{10^n}\right)\]

\[\because n \to +\infty \implies \frac{1}{10^n} \to 0\]

\[\lim_{n\rightarrow+\infty}S_n = \lim_{n\rightarrow+\infty}\frac{1}{3}\left(1 - \frac{1}{10^n}\right) = \frac{1}{3}\]

Formal Definition

If \(S_n\) is the sequence³ of partial sums of the series \(u_1 + u_2 + \ldots u_i + \ldots\) and converges to a limit² \(S\) then it is called the sum¹ of series.

\[S = \sum_{i = 1}^{\infty} u_i\]

However if the sequence diverges then the series is said to diverge and it has no sum.¹

Geometric Series

A series in which each term is a result of previous term being multiplied to a constant, is called geometric series.

\[a + ar + ar^2 + \ldots + ar^{i - 1} + \ldots\]

Theorem

This series converges if \(\lvert r \rvert < 1\) and diverges if \(\lvert r \rvert > 1\)
If the series converges then the sum¹ is

\[\frac {a}{1 - r}\]

Harmonic Series

A series of the form

\[\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\cdots\]

is called harmonic series.

Divergence Test

If \(\lim_{n \to +\infty} u_n \ne 0\) then the series \(\sum u_n\) diverges.
If \(\lim_{n \to +\infty} u_n = 0\) then the series \(\sum u_n\) either converges or diverges.

Properties

If the \(\sum u_i\) and \(\sum v_i\) are convergent series then the following are also convergent
- \[\sum_{i=1}^{\infty}(u_{i}+v_{i})=\sum_{i=1}^{\infty}u_{i}+\sum_{i=1}^{\infty}v_{i}\]
- \[\sum_{i=1}^{\infty}(u_{i}-v_{i})=\sum_{i=1}^{\infty}u_{i}-\sum_{i=1}^{\infty}v_{i}\]
If \(c\) is a constant then both series, \(\sum u_i\) and \(\sum c \cdot u_i\) both either converge or diverge. In case of convergence, we get \(\(\sum_{i=1}^{\infty}c \cdot u_{i}=c\sum_{i=1}^{\infty}u_{i}\)\)
Convergence and Divergence are unaffected if we delete arbitrary amount of terms from the series. That is to say that following both series will not have their convergence or divergence affected
- \[\sum_{i=1}^{\infty}u_{i}=u_{1}+u_{2}+u_{3}+\cdots\]
- \[\sum_{i=K}^{\infty}u_{i}=u_{K}+u_{K+1}+u_{K+2}+\cdots\]

Integral Test

If \(f(x)\) is a decreasing⁴ continuous function⁵ defined over the interval⁶ \([a, +\infty]\) then

\[\sum_{i = 1}^{\infty} u_i\]

\[\int_a^{\infty} f(x) dx\]

Both either converge or diverge.

Hyper Harmonic or P-series

The series of the form

\[\sum_{i=1}^{\infty}\frac{1}{i^{p}}=1+\frac{1}{2^{p}}+\frac{1}{3^{p}}+\cdots\]

where \(p > 0\), is called hyper harmonic or p series.
If \(p > 1\), the series converges.
If \(0 < p \le 1\), then series diverges.