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Chapter 19: Problem 3

Find the sum of the infinite geometric series with first term 2 and common ratio \(\frac{1}{2}\).

Short Answer

Expert verified
Answer: The sum of the infinite geometric series is 4.

Step by step solution

01

Write Down the Formula for the Sum of an Infinite Geometric Series

The formula for the sum of an infinite geometric series is: \(S = \frac{a}{1-r}\)
02

Plug in the Given Values for a and r

The given first term \(a\) is 2, and the common ratio \(r\) is \(\frac{1}{2}\). Plug these values into the formula: \(S = \frac{2}{1 - \frac{1}{2}}\)
03

Simplify the Expression

Now, simplify the expression to find the sum of the series: \(S = \frac{2}{\frac{1}{2}}\)
04

Calculate the Sum of the Series

To find the sum of the series, calculate the fraction: \(S = 2 \cdot 2\) \(S = 4\) The sum of the infinite geometric series with the first term \(2\) and common ratio \(\frac{1}{2}\) is \(4\).

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Most popular questions from this chapter

The sum to infinity of a geometric sequence is four times the first term. Find the common ratio.

A geometric series has \(S_{3}=\frac{37}{8}\) and \(S_{6}=\frac{3367}{512}\). Find the first term and the common ratio.

Find the sum of the first five terms of the geometric sequence with first term 3 and common ratio 2 .

Use the power series expansion of \(\cos x\) to show that $$ \cos \frac{x}{2}=1-\frac{x^{2}}{8}+\frac{x^{4}}{384}-\frac{x^{6}}{46080}+\cdots $$

Find the Maclaurin expansion for \(\sin ^{2} x\). (Hint: use a trigonometrical identity and the series for \(\sin x\).)
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