Arithmetic & Geometric Series

Definitions

Series - A series is formed by the sum or addition of the terms in a sequence. For example an arithmetic series is formed by summing the terms in an arithmetic sequence.

 Sequence: (1) $$3, 5, 7, 9,...$$ Series: (2) $$3+5+7+9+...$$

nth Partial Sum - This is defined as the sum from the 1st term to the nth term in the sequence. For example the 5th partial sum of the series above would be:

(3)
$$3+5+7+9+11=35$$

Notation for Partial Sums

This comes in two flavors. One that is pretty easy to understand and the most commonly used notation in SL math. The second is more complex, scary and used only occasionally in SL.

First notation: An nth partial can be denoted by $S_n$ where n tells the reader how many terms to sum. For example using the arithmetic series $3+5+7+9+...$ then the 5th partial sum would be:

(4)
$$S_5 = 3+5+7+9+11 = 35$$

Its worth noting that the "nth partial sum" generally refers to the actual result of the addition not the statement showing all of the addition.

Second Notation: This notation is often called "summation notation" and includes what term number we start with, what term number we finish with and the details of what we are summing. For example:

This notation allows us to begin summing at any term we would like. The following example starts summing at the 5th term and continues until the 7th term.

The "equation" part of the summation notation works by simple substitution of values for n into the equation, doing the arithmetic results in one of the terms in the series.

Summation of Arithmetic Series

With a bit of cleverness and observation it is possible to create two equations that allow easy calculation for an arithmetic series:

(5)
\begin{align} S_n=\frac{n}{2} (u_1 + u_n) \end{align}

This formula requires you to know the first and last term in the series. A second equation, shown below, allows you to calculate the sum without having to know the final term that you will be summing (clever!). The next equation allows you to calculate the sum if you know the first term and the common difference.

(6)
\begin{align} S_n=\frac{n}{2}(2u_1 +(n-1)d) \end{align}

This equation comes from the previous equation when the substitution of $u_n = u_1 + (n-1)d$ is made and the algebraic express is simplified.

Examples

Example 1 Find the 12th partial sum for $4.5+6+7.5+...$

Example 2 Find $S_{20}$ for $5+2+ -1 + -4$

Example 3 Find the value of n such that $S_n=205$ for $7+10+13+...$

Summation of Geometric Series

Working out an equation for the summation of a geometric series requires a good deal of cleverness and a healthy dose of what feels like magic.

The end result is the formula:

(14)
\begin{align} S_n=\frac{u_1(1-r^n)}{1-r} \end{align}

Where r is the common ratio and $u_1$ is the first term in the series.

Examples

Example 1 Find $S_{20}$ for $\frac{1}{2}+1+2+4+8+...$

Example 2 Find for $18+30+50+...$ find n when $S_n \approx 443,088$

A Few More Definitions

Infinite Series - A series that has a infinite number of terms. I'd give an example but it would take a while to write down…

Divergent Infinite Series- A series that when all the infinite number of terms are added the result is positive or negative infinity. If this seems like, "Duh, doesn't that always happen?" take a peek at the next definition.

Convergent Infinite Series - A series that when all the infinite number of terms are added the result is a finite (or non-infinite) number. How's that possible you ask? Try adding up the series below on your calculator.

(19)
\begin{align} 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+... \end{align}

Add more and more terms in this series and it will "converge" or get closer and closer to 2 but will never get there, because you'd have to literally add an infinite number of terms. Or try:

(20)
\begin{align} 3-\frac{4}{2 \times 3 \times 4}+\frac{4}{4 \times 5 \times 6}-\frac{4}{6 \times 7 \times 8}+\frac{4}{8 \times 9 \times 10}-... = \pi \end{align}

While it may seem silly and useless to talk about the sum of an infinite number of terms, but there are some "interesting" and useful results.

Infinite Geometric Series

It terms out that (infinite) geometric series will converge if $-1< r <1$. Looking back at the equation for the sum of a geometric series:

(21)
\begin{align} S_n = \frac{u_1 (1-r^n)}{1-r} \end{align}

If $-1< r <1$ and n gets bigger and bigger then $r^n$ will get closer and closer to zero. So "when n gets to infinity" (it can't but its easier to say this way) then $r^{\infty} = 0$ leave us with the equation:

(22)
\begin{align} S_\infty = \frac{u_1}{1-r} \end{align}

Nice. And. Simple.

Example

Example 1 Find the value of the infinite sum of $1+\frac{2}{3} +\frac{4}{9}+...$