Finding the Sum of an Infinite Series

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Question

Calculate the sum of the following series:

{\displaystyle {\begin{aligned}2+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}+\ldots \end{aligned}}}

Solution

Step (1): Express in Sigma Notation

The first four terms in the series are

{\displaystyle {\begin{aligned}2&=2^{1}\\{\frac {1}{2}}&=2^{-1}\\{\frac {1}{8}}&=2^{-3}\\{\frac {1}{32}}&=2^{-5}\\\end{aligned}}}

Each term in the series is the same multiple (${\displaystyle 1/4}$) of the previous term. Hence, the series is a geometric series with common ratio ${\displaystyle r=1/4}$ and first term ${\displaystyle a=2}$:

{\displaystyle {\begin{aligned}2+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}+\ldots =\sum _{k=1}^{\infty }ar^{k-1}=\sum _{k=1}^{\infty }2{\Big (}{\frac {1}{4}}{\Big )}^{k-1}\end{aligned}}}

Step (2): Apply Summation Formula

{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }2{{\Bigl (}{\frac {1}{4}}{\Bigr )}}^{k-1}={\frac {2}{1-1/4}}={\frac {8}{3}}\end{aligned}}}

Explanation of Each Step

Step (1)

In any question where one must find the sum of a series given in the form

{\displaystyle {\begin{aligned}a_{1}+a_{2}+a_{3}+\ldots \end{aligned}}}

where each term is positive, we must first convert the sum to sigma notation. Why? Because there are no methods (covered in the ISM) to compute an infinite sum otherwise.

There are no general methods to do this, but by looking for a patterns, one might want to look for a way to relate each term by a common base. In our example here, we found that each term in the series could be related to each other with a common ratio of 1/4.

Step (2)

Applying our summation formula with common ratio ${\displaystyle 1/4}$ and first term ${\displaystyle a=3}$ yields Step (2).

Possible Challenges

Getting Started

The question asks us to compute the sum of an infinite series, and there are only two ways we could do this. The only two series that have methods for which we can calculate their sums are geometric and telescoping. Since each term is positive, the sum is not telescoping. So the series must be geometric, which means we can get started by looking for a common ratio, ${\displaystyle r}$.

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Next: A Geometric Series Problem with Shifting Indicies