Use the formula for the sum of a geometric series to find the sum or state that the series...

Question:

Use the formula for the sum of a geometric series to find the sum or state that the series diverges.

{eq}\displaystyle \sum_{n=6}^{\infty} \frac{3^n}{14^n} {/eq}

Convergent Infinite Geometric Series:

Infinite geometric series are series of the form {eq}\displaystyle \sum_{n=0}^{\infty} ar^n {/eq} where {eq}a {/eq} pertains to its first term and {eq}r {/eq} refers to its ratio.

Infinite geometric series converges to the value {eq}\displaystyle \frac{a}{1-r} {/eq} if the absolute value of {eq}r {/eq} is less than {eq}1 {/eq}.

Answer and Explanation:

We show first that the given sum {eq}\displaystyle \sum_{n=6}^{\infty} \frac{3^n}{14^n} {/eq} is, indeed, a convergent infinite geometric series:

{e...

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Learn more about this topic:

Understand the Formula for Infinite Geometric Series

from Algebra II Textbook

Chapter 21 / Lesson 11
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