Week 2 – precalculus 11

Infinite Geometric Series

This week we learned how to identify an infinite geometric series and how they work. An infinite geometric series is a sequence of numbers that are being multiplied by the same amount each consecutive time in the sequence and all the numbers are being added together. The common ratio, r, is the term used for the amount being multiplied each time. There are a some rules for a converging infinite geometric series, one being that r must be above -1 or below 1. This causes each term to lower in the sequence. This means that in an infinite series that follows this rule for r, the sum will eventually plateau because the number being added is so small. You are multiplying by a decimal so the terms are lower each time. This means that you can find the sum of infinity because the numbers being added are so small that they are not changing the number at all. If the common ratio is above 1 or below -1, it becomes a diverging infinite geometric series. We cannot find the sum of this because each time a number is added the sum of all numbers is changing drastically.

{t_1} is the first term in the sequence. ‘r’ represents the common ratio in the sequence. ‘n’ is the number we are adding until in the series.

To find the sum of an converging geometric series that is infinite we use the equation: S_{\propto} = \frac {t_1}{1-r}

The way we get this formula is from the already existing formula of a finite geometric series, this formula adds up every number in the series until n is reached: S_{n} = \frac {t_1 (1 - r^n)} {1 - r}

When r is below 1 and above -1 r^{n} gets closer and closer to zero because n is getting bigger. A fraction multiplied by itself creates a smaller fraction each time.  This changes the equation and makes S_{n}: \frac {t_1 (1 - 0)} {1 - r}

This creates the equation: S_{\propto} = \frac {t_1}{1-r} because $latex r^{n} is now 0

 

An example of this in action would be the following:

r=\frac {1}{2} and t_1 = 10

We use S_{\propto} = \frac {t_1}{1-r}

Substitute: S_{\propto} = \frac {10}{1-\frac {1}{2}}

S_{\propto} = \frac {10}{\frac {1}{2}}

Then:

S_{\propto} = 20

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