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Chapter 8: Problem 52

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$\frac{1}{9}+\frac{2}{9^{2}}+\frac{3}{9^{3}}+\dots+\frac{n}{9^{n}}$$

Short Answer

Expert verified
The summation notation of the series is \(\sum_{i=1}^{n}\frac{i}{9^{i}}\).

Step by step solution

01

Understand the Series

First, it's important to recognize the structure of the series. The series goes from 1 to n, each term of the series is represented as \(\frac{i}{9^{i}}\), where i indicates the term number.
02

Determine the Summation Structure

Rewrite each term as \(\frac{i}{9^{i}}\), for i ranging from 1 to n.
03

Express in Summation Notation

The Sigma notation for this series will be \(\sum_{i=1}^{n}\frac{i}{9^{i}}\). The sum of the series is the sum of the terms when each term follows this rule.

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Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.

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