24.2: Infinite Geometric Series
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In some cases, it makes sense to add not only finitely many terms of a geometric sequence, but all infinitely many terms of the sequence! An informal and very intuitive infinite geometric series is exhibited in the next example.
Consider the geometric sequence 1,12,14,18,116,…
Solution
Here, the common ratio is r=12, and the first term is a1=1, so that the formula for an is an=(12)n−1. We are interested in summing all infinitely many terms of this sequence:
1+12+14+18+116+…
We add these terms one by one, and picture these sums on the number line:
1=11+12=1.51+12+14=1.751+12+14+18=1.8751+12+14+18+116=1.9375
We see that adding each term takes the sum closer and closer to the number 2. More precisely, adding a term an to the partial sum a1+⋯+an−1 cuts the distance between 2 and a1+⋯+an−1 in half. For this reason we can, in fact, get arbitrarily close to 2, so that it is reasonable to expect that
1+12+14+18+116+⋯=2
In the next definition and observation, this equation will be justified and made more precise. We start by providing the definition of an infinite series.
An infinite series is given by the
∞∑i=1ai=a1+a2+a3+…
To be more precise, the infinite sum is defined as the limit ∞∑i=1ai:=lim. Therefore, an infinite sum is defined, precisely when this limit exists.
Let \{a_n\} be a geometric sequence with a_n=a_1\cdot r^{n-1}. Then the infinite geometric series is defined whenever -1<r<1. In this case, we have:
\label{EQU:inf-geo-series} \boxed{ \sum_{i=1}^\infty a_i = a_1\cdot \dfrac{1}{1-r} }
- Proof
-
Informally, this follows from the formula \sum_{i=1}^k a_i = a_1\cdot \dfrac{1-r^{k}}{1-r} and the fact that r^k approaches zero when k increases without bound.
More formally, the proof uses the notion of limits, and goes as follows:
\sum_{i=1}^\infty a_i = \lim_{k\to \infty}\bigg( \sum_{i=1}^k a_i \bigg) = \lim_{k\to \infty}\bigg( a_1\cdot \frac{1-r^{k}}{1-r}\bigg) = a_1\cdot \frac{1-\lim\limits_{k\to\infty } (r^{k})}{1-r} = a_1\cdot \dfrac{1}{1-r} \nonumber
Find the value of the infinite geometric series.
- \sum_{j=1}^\infty a_j, for a_j=5\cdot \left(\dfrac 1 3\right)^{j-1}
- \sum_{n=1}^\infty 3\cdot \left(0.71\right)^n
- 500-100+20-4+\dots
- 3+6+12+24+48+\dots
Solution
- We use formula \ref{EQU:inf-geo-series} for the geometric series a_n=5\cdot \left(\dfrac 1 3\right)^{n-1}, that is a_1=5\cdot \left(\dfrac 1 3\right)^{1-1}=5\cdot \left(\dfrac 1 3\right)^{0}=5\cdot 1 =5 and r=\dfrac 1 3. Therefore,
\sum_{j=1}^\infty a_j=a_1\cdot \dfrac {1}{1-r}=5\cdot \dfrac {1}{1-\frac 1 3}=5\cdot \dfrac 1 {\frac{3-1}{3}}=5\cdot \dfrac 1 {\frac 2 3}=5\cdot \dfrac 3 2 =\dfrac {15} 2 \nonumber
- In this case, a_n=3\cdot (0.71)^n, so that a_1=3\cdot 0.71^1=3\cdot 0.71=2.13 and r=0.71. Using again formula \ref{EQU:inf-geo-series}, we can find the infinite geometric series as
\sum_{n=1}^\infty 3\cdot \left(0.71\right)^n=a_1\cdot \dfrac{1}{1-r}=2.13\cdot \dfrac 1 {1-0.71}=2.13\cdot \dfrac 1 {0.29}=\dfrac {2.13}{0.29}=\dfrac{213}{29} \nonumber
In the last step we simplified the fraction by multiplying both numerator and denominator by 100, which had the effect of eliminating the decimals.
- Our first task is to identify the given sequence as an infinite geometric sequence:
\{a_n\} \text{ is given by } 500, -100, 20, -4, \dots \nonumber
Notice that the first term is 500, and each consecutive term is given by dividing by -5, or in other words, by multiplying by the common ratio r=-\dfrac 1 5. Therefore, this is an infinite geometric series, which can be evaluated as
\begin{aligned} 500-100+20-4+\dots &= \sum_{n=1}^\infty a_n\\&=a_1\cdot \dfrac 1 {1-r}\\&=500\cdot \dfrac 1 {1-\left(-\frac 1 5\right)} \\ &= 500 \cdot \dfrac{1}{1+\frac 1 5} \\&=\dfrac {500}{\frac {1+5}{5}}\\&=\dfrac{500}{\frac 6 5}\\&=500\cdot \dfrac 5 6 \\ &= \dfrac {2500}{6} \\&= \dfrac{1250}{3}\end{aligned} \nonumber
- We want to evaluate the infinite series 3+6+12+24+48+\dots. The sequence 3, 6, 12, 24, 48, \dots is a geometric sequence, with a_1=3 and common ratio r=2. Since r\geq 1, we see that formula \ref{EQU:inf-geo-series} cannot be applied, as \ref{EQU:inf-geo-series} only applies to -1<r<1. However, since we add larger and larger terms, the series gets larger than any possible bound, so that the whole sum becomes infinite.
3+6+12+24+48+\dots=\infty \nonumber
The fraction 0.55555\dots may be written as:
0.55555\dots = 0.5+0.05+0.005+0.0005+0.00005+\dots \nonumber
Noting that the sequence
0.5,\underset{\times 0.1}{\hookrightarrow } 0.05, \underset{\times 0.1}{\hookrightarrow } 0.005, \underset{\times 0.1}{\hookrightarrow } 0.0005, \underset{\times 0.1}{\hookrightarrow } 0.00005, \dots \nonumber
is a geometric sequence with a_1=0.5 and r=0.1, we can calculate the infinite sum as:
0.55555\dots = \sum_{i=1}^\infty 0.5 \cdot \left(0.1\right)^{i-1}= 0.5\cdot \dfrac{1}{1-0.1} = 0.5 \cdot\dfrac{1}{0.9}=\dfrac{0.5}{0.9}=\dfrac 5 9 \nonumber
Here we multiplied numerator and denominator by 10 in the last step in order to eliminate the decimals.