Question

Find the sum of the following series. If it is divergent, type "Diverges" or "D". [ -sum_{n=1}^{infty} frac{1}{n}left(-frac{1}{2} ight)^{n}=frac{1}{2}-frac{1}{8}+frac{1}{24}-frac{1}{64}+cdots ] Answer: Note: You cannot write a decimal number for the answer. Hint: What power series is equal to this sum?

          Find the sum of the following series. If it is divergent, type "Diverges" or "D".
[
-sum_{n=1}^{infty} frac{1}{n}left(-frac{1}{2}
ight)^{n}=frac{1}{2}-frac{1}{8}+frac{1}{24}-frac{1}{64}+cdots
]
Answer:
Note: You cannot write a decimal number for the answer.
Hint: What power series is equal to this sum?

$Find the sum of the following series. If it is divergent, type "Diverges" or "D". [ -sumn=1^infty frac1nleft(-frac12 ight)^n=frac12-frac18+frac124-frac164+cdots ] Answer: Note: You cannot write a decimal number for the answer. Hint: What power series is equal to this sum?$

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Calculus

Ron Larson, Robert P. Hostetler, Bruce… 8th Edition

Chapter 9

Instant Answer

Solved by Expert Wei Zhang

06/19/2023

Step 1

Let's consider the geometric series: \[ \sum_{n=0}^{\infty} \left(-\frac{1}{2}\right)^n \] Using the geometric series formula, we have: \[ \sum_{n=0}^{\infty} \left(-\frac{1}{2}\right)^n = \frac{1}{1 - (-\frac{1}{2})} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \] Show more…

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