Question

Write out each sum. $$ \sum_{k=0}^{n} \frac{1}{3^{k}} $$

   Write out each sum.
$$
\sum_{k=0}^{n} \frac{1}{3^{k}}
$$

Algebra and Trigonometry

Algebra and Trigonometry

Michael Sullivan 10th Edition

Chapter 13, Problem 53

Instant Answer

Solved by Expert Julie Silva

06/21/2020

Step 1

We do this by substituting 0 in place of k in the given expression. So, we have $\frac{1}{3^{0}}$. Since any number to the power of 0 is 1, we have $\frac{1}{1}$, which simplifies to 1. So, the first term in the sequence is 1. Show more…

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Write out each sum. $$ \sum_{k=0}^{n} \frac{1}{3^{k}} $$

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Key Concepts

Summation Notation

Summation notation uses the sigma symbol to indicate the sum of a sequence of terms. It includes details such as the index of summation, lower and upper bounds, and the general term, providing a compact and systematic way to represent a series of additions.

Geometric Series

A geometric series is one in which each term is a constant multiple of the previous term. Recognizing a summation as a geometric series is valuable because it allows the use of specific formulas for the sum of the series, both for finite and infinite cases.

Exponential Functions

Exponential functions involve expressions where the variable appears in the exponent. In the context of the summation, the term 1/3^k demonstrates an exponential decay factor that characterizes how each term scales, which is central to understanding its geometric progression.

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