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Infinite Geometric Series

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Infinite Geometric Series

Study Guide

Key Definition

An infinite geometric series is the sum of an infinite geometric sequence. The general form is $a_1 + a_1 r + a_1 r^2 + a_1 r^3 + \ldots$, where $a_1$ is the first term and $r$ is the common ratio.

Important Notes

When $\lvert r \rvert < 1$, the series converges.
The sum $S$ of a convergent infinite geometric series is $S = \frac{a_1}{1 - r}$.
If $r \geq 1$, the series does not converge.
The series is infinite because it has no last term.
Convergence depends on the value of $r$.

Mathematical Notation

$\sum$ represents the sum of a series

$\infty$ denotes infinity

$| \cdot |$ represents the absolute value

$\frac{a_1}{1 - r}$ is the formula for the sum of a convergent series

Remember to use proper notation when solving problems

Why It Works

The formula $S = \frac{a_1}{1 - r}$ works because as the number of terms approaches infinity, the terms become negligible if $| r | < 1$.

Remember

For convergence, ensure $| r | < 1$.

Quick Reference

Convergence Condition:$| r | < 1$

Sum Formula:$S = \frac{a_1}{1 - r}$

Understanding Infinite Geometric Series

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A simple infinite geometric series can be written as $a_1 + a_1 r + a_1 r^2 + \ldots$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the infinite series $5 + 5 \left(\frac{1}{2}\right) + 5 \left(\frac{1}{2}\right)^2 + \ldots$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A ball is dropped from a height of $10 \text{ meters}$ and bounces back to $\frac{3}{4}$ of its previous height each time. What is the total distance it travels?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider an infinite series where each term is half of the previous term starting from $8$. What is the sum?

Challenge Quiz

Single Choice Quiz

Advanced

Find the sum of the infinite series $3 - 3 \left(\frac{1}{3}\right) + 3 \left(\frac{1}{3}\right)^2 - \ldots$.

Recap

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