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Chapter 9: Problem 116

Find the indicated partial sum of the series. \(\sum_{n=1}^{\infty} 8\left(-\frac{1}{4}\right)^{n}\) Fourth partial sum

Short Answer

Expert verified
After doing the calculations, the fourth partial sum of the series is \(\frac{224}{21}\).

Step by step solution

01

Identify the terms in the series

Here, a geometric series is given. The first term of the series \(a = 8*(-\frac{1}{4})\) and the common ratio \(r = -\frac{1}{4}\)
02

Use the formula for the sum of a geometric series

The formula for the sum of a geometric series is \(S = \frac{a(1-r^n)}{1-r}\) where \(S\) is the sum of the first \(n\) terms, \(a\) is the first term, \(r\) is the common ratio and \(n\) is the number of terms.
03

Calculate the fourth partial sum using the formula

Substitute \(a = 8*(-\frac{1}{4})\), \(r = -\frac{1}{4}\), and \(n = 4\) into the formula: \(S = \frac{a(1-r^n)}{1-r} = \frac{8*(-\frac{1}{4})*(1-(-\frac{1}{4})^4)}{1-(-\frac{1}{4})}\)

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