Question

Find each sum. $$\sum_{k=1}^{\infty}(0.1)^{k}$$

   Find each sum.
$$\sum_{k=1}^{\infty}(0.1)^{k}$$

A Graphical Approach to Precalculus with Limits

A Graphical Approach to Precalculus with Limits

John Hornsby,… 7th Edition

Chapter 12, Problem 64

Instant Answer

Solved by Expert Uma Kumari

08/19/2020

Step 1

The general form of a geometric series is $\sum_{k=0}^{\infty} ar^{k}$, where $a$ is the first term and $r$ is the common ratio. Show more…

Show all steps

lock

Thanks for your feedback!

Find each sum. $$\sum_{k=1}^{\infty}(0.1)^{k}$$

Play audio

Feedback

Powered by NumerAI

Uma Kumari and 51 other subject Calculus 2 / BC educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Infinite Series

An infinite series is the sum of an infinite sequence of terms. Understanding infinite series involves studying how the sums of these terms behave as more and more terms are added, and whether the overall sum converges to a particular value or diverges, which is fundamental to many areas in calculus and analysis.

Geometric Series

A geometric series is a type of infinite series where each term after the first is found by multiplying the previous term by a constant called the common ratio. This concept simplifies the process of summing the series as it allows the derivation of a straightforward formula to determine the sum if it converges.

Convergence of Infinite Geometric Series

Convergence in the context of an infinite geometric series refers to the condition under which the infinite sum approaches a finite limit. Specifically, an infinite geometric series converges if and only if the absolute value of the common ratio is less than one. This principle is essential when determining whether an infinite series has a finite sum.

Summation Formula for Infinite Geometric Series

The summation formula for an infinite geometric series provides a direct method to compute its sum when the series converges. If the first term is 'a' and the common ratio is 'r' (with |r| < 1), the sum of the series is given by a/(1-r). This formula is a critical tool in both theoretical and applied mathematics when working with series that exhibit constant factor progression.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later