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Chapter 11: Problem 50

Find the sum, if it exists. $$\sum_{k=1}^{\infty} \frac{8}{3}\left(\frac{1}{2}\right)^{k-1}$$

Short Answer

Expert verified
The sum of the series is \(\frac{16}{3}.)\)

Step by step solution

01

Identify the Series Type

The series given is \(\frac{8}{3} \sum_{k=1}^{\infty} \left( \frac{1}{2} \right)^{k-1}.)\) This is a geometric series because the terms involve a common ratio raised to successive powers.
02

Geometric Series Formula

Recall the formula for the sum of an infinite geometric series: \(S = \frac{a}{1-r}.)\) Here, \(a = \frac{8}{3}.)\) Identify \(r = \frac{1}{2}.)\)
03

Check Convergence

A geometric series converges if the absolute value of the common ratio \(|r| < 1.)\) In this case, \(|\frac{1}{2}| = \frac{1}{2} < 1.)\) Therefore, the series converges.
04

Substitute Values into the Formula

Substitute \(a = \frac{8}{3} )\)and \(r = \frac{1}{2}\)into the formula: \(S = \frac{\frac{8}{3}}{1 - \frac{1}{2}}.)\)
05

Simplify the Expression

Perform the arithmetic operations: \(S = \frac{\frac{8}{3}}{\frac{1}{2}} .\) Simplify it to get \(S = \frac{8}{3} \times 2 = \frac{16}{3}.)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Series
When we talk about geometric series, we refer to a sequence where each term is obtained by multiplying the previous term by a constant. This constant is called the common ratio.
A geometric series can be finite or infinite, depending on the number of terms involved.
Our example is an infinite geometric series because it continues indefinitely.
Common Ratio
The common ratio is a key element in geometric series. It determines how each term in the series relates to the previous one.
In mathematical terms, if the first term of a geometric series is 'a' and the common ratio is 'r', the series can be written as:
Series Convergence
Series convergence is an important concept for determining whether the sum of an infinite series is finite.
A series is said to converge if the absolute value of the common ratio is less than 1.
In our example, the common ratio is \(\frac{1}{2} < 1\), so the series converges.
Sum Formula
Once you know that a geometric series converges, you can use the sum formula to find its sum.
The sum formula for an infinite geometric series is:
\(S = \frac{a}{1-r}\)
where 'a' is the first term and 'r' is the common ratio.
In our given series:
\(a = \frac{8}{3}\) and \(r = \frac{1}{2}\)
Substituting these value into the sum formula we get:
\(S = \frac{\frac{8}{3}}{1 - \frac{1}{2}} = \frac{\frac{8}{3}}{\frac{1}{2}} = \frac{16}{3}\)

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