Problem 52 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: Determine the Summation Structure

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

Step 3: 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.