Geometric Series
A Geometric Series is the sum of the values in a Geometric Sequence.
Finite Geometric Series: The first geometric series we will evaluate is a finite geometric series. A finite series as a set number values that are added. We denote the sum of the first \(n\) terms of a geometric series using the notation \(S_n\).
A Geometric Series is the sum of the values in a Geometric Sequence.
Finite Geometric Series: The first geometric series we will evaluate is a finite geometric series. A finite series as a set number values that are added. We denote the sum of the first \(n\) terms of a geometric series using the notation \(S_n\).
Sum of a Finite Geometric Series: \(S_n=a_1\left(\frac{1-r^n}{1-r}\right)\) where \(r\neq0\) and \(r\neq1\)
\(a_1\) is the value of the first term, \(r\) is the common ratio and \(n\) is the number of terms in the series.
\(a_1\) is the value of the first term, \(r\) is the common ratio and \(n\) is the number of terms in the series.
Example 1: Find the sum of the geometric series: \(\sum\limits_{k=1}^{12}3\left(-2\right)^{n-1}\).
\(\begin{align}& r=-2, n=12, a_1=3\ & \ \ \ &\text{1) Identify the r, n and}\ a_1.\\
&S_n=3\left(\frac{1-\left(-2\right)^{12}}{1-\left(-2\right)}\right)\ & \ \ \ &\text{2) Substitute into the formula.}\\
&S_n=3\left(\frac{1-4096}{1+2}\right)\ & \ \ \ &\text{3) Evaluate the exponents.}\\
&S_n=3\left(\frac{-4095}{3}\right)\ & \ \ \ &\text{4) Add/Subtract like terms.}\\
&S_n=-1365\ & \ \ \ &\text{5) Simplify.}\end{align}\)
Example 2: Find the sum of the geometric series \(4+2+1+...+\frac{1}{16}\).
Infinite Geometric Series:
The second geometric series we will evaluate is an infinite geometric series. An infinite series has an infinite number of terms and is denoted with just \(S\).
Consider the two series below. Which of the two would have an infinite sum? What conditions are necessary for an infinite geometric series to have a sum?
Series A: \(1 + 2 + 4 + 8 +….\) Series B: \(8 + 4 + 2 + 1 +....\)
Series A does not have a sum because you will continually add ever larger numbers. Series B will have a sum because the numbers you will be adding get closer and closer to zero.
We can use partial sums to explore the sum of an infinite series. A partial sum means that we are finding the sum of the first term, the sum of the first two terms, the sum of the first three terms, etc. until we see if there is one value that the series will converge upon.
Looking at Series B above, find the following partial sums:
\(S_1=8\)
\(S_2=8+4=12\)
\(S_3=8+4+2=14\)
\(S_4=8+4+2+1=15\)
\(S_5=8+4+2+1+\frac{1}{2}=15.5\)
\(S_6=8+4+2+1+\frac{1}{2}+\frac{1}{4}=15.75\)
Quick Check
Find a few more partial sums. What value do you think the series will converge to?
Quick Check Solutions
Find a few more partial sums. What value do you think the series will converge to?
Quick Check Solutions
Sum of an Infinite Geometric Series: \(S=\large\frac{a_1}{1-r}\)
\(a_1\) is the value of the first term, \(r\) is the common ratio. An infinite sum exists if \(\left|r\right|<1\). We say the series converges. If \(\left|r\right|\ge1\) we say the series diverges and no sum exists. It is necessary to first verify if an infinite sum does, indeed, exist.
\(a_1\) is the value of the first term, \(r\) is the common ratio. An infinite sum exists if \(\left|r\right|<1\). We say the series converges. If \(\left|r\right|\ge1\) we say the series diverges and no sum exists. It is necessary to first verify if an infinite sum does, indeed, exist.
Example 3: Using the formula, find the sum of the infinite geometric series \(8 + 4 + 2 + 1 + ….\)
\(S=\frac{8}{1-\frac{1}{2}}\)
\(S=8(2)\)
\(S=16\)
Example 4: A ball is dropped from an initial height of \(4\) feet. It rebounds to \(75\%\) of its initial height after each bounce. When the ball finally stops, what total distance did it travel?