# Infinite geometric series problem

• MillerGenuine
In summary, an infinite geometric series is a sum of terms where each term is multiplied by a common ratio, with no specific end point. The sum of an infinite geometric series can be calculated using the formula S = a / (1-r), but only if the common ratio is between -1 and 1. An infinite geometric series can have a negative common ratio, resulting in alternating positive and negative terms. The difference between a finite and infinite geometric series is that a finite series has a specific number of terms and a formula for calculating the sum, while an infinite series may not have a finite value. Infinite geometric series are commonly used in various real-life scenarios and in mathematical and scientific fields for analysis and prediction.
MillerGenuine

## Homework Statement

$$\sum_{n=1}^\infty \frac{(-3)^{n-1}}{4^n}$$

## The Attempt at a Solution

$$\sum_{n=1}^\infty \frac{(-3)^n-1}{4^n}$$

$$\frac{1}{4}\sum_{n=1}^\infty \frac(-{3}{4})^{n-1}$$

Can some one please explain how they got from the first step to the 2nd. How do you pull out a 1/4 and how does the "n" on the 4 dissapear?
Thanks

Can you fix your third summation? It's all fouled up.

## What is an infinite geometric series?

An infinite geometric series is a sum of terms where each term is multiplied by a common ratio. The series continues indefinitely, with no specific end point.

## How do I find the sum of an infinite geometric series?

The sum of an infinite geometric series can be calculated using the formula S = a / (1-r), where S is the sum, a is the first term, and r is the common ratio. However, this formula only works if the common ratio is between -1 and 1. If the common ratio is outside of this range, the series will either diverge or oscillate.

## Can an infinite geometric series have a negative common ratio?

Yes, an infinite geometric series can have a negative common ratio. This will result in alternating positive and negative terms in the series.

## What is the difference between a finite and infinite geometric series?

A finite geometric series has a specific number of terms, while an infinite geometric series has an infinite number of terms. Additionally, the sum of a finite geometric series can be calculated using a formula, while the sum of an infinite geometric series may not have a finite value.

## How is an infinite geometric series used in real life?

Infinite geometric series can be used to model real-life scenarios such as population growth, compound interest, and radioactive decay. They can also be used in various mathematical and scientific fields to analyze patterns and make predictions.

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