As we’ve already seen, using Summation Notation, also called Series Notation, enables us to add up the terms of a sequence by creating Partial Sums.
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
But wouldn’t it be nice if we didn’t have to add up all those terms? If only there was a formula that we could just plug into!
Well, happy day! Because this lesson is all about two very special types of Series: Arithmetic and Geometric Series where all we have to do to is plug into a formula!
Super simple and super easy!
Now, remember, and Arithmetic Sequence is one where each term is found by adding a common value to each term and a Geometric Sequence is found by multiplying a fixed number to each term. Thus making both of these sequences easy to use, and allowing us to generate a formula that will enable us to find the sum in just a few simple steps.
We will begin by exploring the Arithmetic Series and it’s Summation Formula. What is extremely important to note, and should be a warning to us, is that we can only find the sum of an Arithmetic Series that is Finite! That means, we can only find the sum for the first n terms.
We start by using the Arithmetic Series formula to find the sum of various Arithmetic Series, and then we will work backwards, from our Sum and locate the first term and the common difference.
Formula for Finding the Sum of a Geometric Series
Next, we will look at the formula for a Finite Geometric Series, and how to use it to find the sum of the first n terms of a Geometric sequence.
Then, we will spend the rest of the lesson discussing the Infinite Geometric Series.
This series is so special because it will enable us to find such things as Power Series and Power Functions in Calculus!
Once again, there is a warning. We will only be able to find the sum of an Infinite Geometric Series under certain conditions, or as Purple Mathsays “special circumstance.” That is to say that the infinite series will only converge (i.e., be able to find the sum) if and only if the ratio r is between –1 and 1.
So, we will take the time to discuss how we can even find the sum of an infinite series, and see why/how it works, and then use it to find the sum of various infinite geometric series.
Geometric Series – Video
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