Finite and Infinite Geometric Series

Introduction to Geometric Progressions

In the TMUA and ESAT, you are expected to recognise geometric series, often referred to as Geometric Progressions (GPs). A geometric progression is defined by its first term $a$ and its common ratio $r$. Each term $u_n$ is related to the previous term by the ratio $u_{n+1} / u_n = r$. This leads to the standard formula for the $n^{th}$ term:

$u_n = ar^{n-1}$

Note that sometimes the first term is written as $ar^0$ to remain consistent with this formula.

The Formulas for Series Summation

A geometric series is the sum of the terms of a geometric progression. For a finite number of terms $n$, the sum $S_n$ is given by:

$S_n = \frac{a(1 - r^n)}{1 - r}$

For an infinite geometric series, a sum only exists if the series is convergent. This happens when the common ratio $r$ satisfies the condition $|r| < 1$. The sum to infinity, $S_\infty$, is given by:

$S_\infty = \frac{a}{1 - r}$

It is also essential to be fluent in using Sigma ($\Sigma$) notation to represent these sums. The finite sum can be written in two equivalent ways:

$S_n = \sum_{k=1}^{n} ar^{k-1} = \sum_{k=0}^{n-1} ar^k$

Generating New Progressions

You can derive new progressions from a given geometric progression. For example, given the series $S_n = a + ar + ar^2 + ar^3 + ar^4 + ar^5 + \dots$, replacing $r$ with $-r$ creates an alternating series:

$S_n(-r) = a - ar + ar^2 - ar^3 + ar^4 - ar^5 + \dots$

This is also a GP with first term $a$ and common ratio $-r$. Its sum to infinity is $S_\infty = \frac{a}{1+r}$. By combining these, you can explore other series. For instance, $\frac{1}{2}(S_n + S_n(-r)) = a + ar^2 + ar^4 + ar^6 + \dots$, which is a GP with first term $a$ and common ratio $r^2$.

Example 1: Squaring every term

Consider a series where every term is squared:

$S_n(\text{squared}) = a^2 + a^2r^2 + a^2r^4 + a^2r^6 + \dots$

This is a GP where the new first term is $u_1 = a^2$ and the new common ratio is $r^2$. The sum to infinity for this squared series is $S_\infty = \frac{a^2}{1 - r^2}$.

Example 2: Raising every term to power $k$

If every term is raised to a positive integer power $k$:

$S_n(\text{power } k) = a^k + a^k r^k + a^k r^{2k} + a^k r^{3k} + \dots$

This is a GP with first term $a^k$ and common ratio $r^k$. If $|r| < 1$ and $k$ is a positive number, then $|r^k| < 1$, and a sum to infinity will exist. However, if $k < 0$ and $0 < r < 1$, the ratio $r^k$ will be greater than 1, meaning the series will no longer converge.

Summing a Subset of a Geometric Series

You may be asked to find the sum of part of a GP, such as $ar^m + ar^{m+1} + \dots + ar^n$, where $n > m$.

One critical concept here is the number of terms. It is common to assume there are $n - m$ terms, but this is a fence post error. There are actually $n - m + 1$ terms in this sum.

There are three main methods for tackling such a sum:

1. Method 1: Difference of two sums. Treat it as $S_{n+1} - S_m$. Note that $S_{n+1}$ includes terms up to $ar^n$ and $S_m$ removes terms up to $ar^{m-1}$, leaving the required terms.

2. Method 2: Factorise $r^m$ or $ar^m$. Factorising $ar^m$ gives $ar^m(1 + r + r^2 + \dots + r^{n-m})$. This is a standard GP inside the brackets with $n - m + 1$ terms.

3. Method 3: New GP definition. Treat it as a new GP where the first term is $ar^m$, the common ratio is $r$, and the number of terms is $n - m + 1$.

As an exercise, you should work out the sum using all three methods to verify they produce the same result.