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.
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.
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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CalcWorkshop has been a crucial resource in helping me pass my Discrete Mathematics course. The tutorials were well-structured and broke down difficult topics into manageable sections. I appreciated the detailed explanations and the practice problems that reinforced what I learned. This site truly made a difference in my understanding and performance in class. The value I received from CalcWorkshop was well worth the price. It's an investment in your education that pays off.
Very great resource for students who need the extra help for their math courses. Videos are super helpful and in depth. Lots of practice problems and tests for mastery.
I really enjoyed and got a lot out of the online videos. I wanted a comprehensive Calculus 3 review that gave plenty of practice problems and explained the concepts in a fun and engaging way. This delivered!
Jen's Discrete Math Workshop: A Clear Path to SuccessI can't recommend Jen's workshop enough! Before attending, I felt a bit overwhelmed by the sheer scope of discrete math. Jen's thoughtful outline of the subject provided a much-needed overview, helping me understand the big picture and how each topic fits in.Each lecture is filled with clear explanations and practical examples, making even complex concepts easier to grasp. Jen's passion for the subject shines through, and her dedication to helping students truly understand the material is evident.Thanks to Jen, I'm finally enjoying discrete math! If you're looking for a workshop that will leave you feeling confident and excited about the subject, look no further.
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I decided to subscribe about 12 weeks into the 16 week Calculus I class. So far it has helped tremendously. I wish I had signed up at week 1. I look forward to learning with Jenn and the Calcworkshop for future classes.Thank you, Jenn!Regards,Frank
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CalcWorkshop has been a crucial resource in helping me pass my Discrete Mathematics course. The tutorials were well-structured and broke down difficult topics into manageable sections. I appreciated the detailed explanations and the practice problems that reinforced what I learned. This site truly made a difference in my understanding and performance in class. The value I received from CalcWorkshop was well worth the price. It's an investment in your education that pays off.
Very great resource for students who need the extra help for their math courses. Videos are super helpful and in depth. Lots of practice problems and tests for mastery.
I really enjoyed and got a lot out of the online videos. I wanted a comprehensive Calculus 3 review that gave plenty of practice problems and explained the concepts in a fun and engaging way. This delivered!
Jen's Discrete Math Workshop: A Clear Path to SuccessI can't recommend Jen's workshop enough! Before attending, I felt a bit overwhelmed by the sheer scope of discrete math. Jen's thoughtful outline of the subject provided a much-needed overview, helping me understand the big picture and how each topic fits in.Each lecture is filled with clear explanations and practical examples, making even complex concepts easier to grasp. Jen's passion for the subject shines through, and her dedication to helping students truly understand the material is evident.Thanks to Jen, I'm finally enjoying discrete math! If you're looking for a workshop that will leave you feeling confident and excited about the subject, look no further.
Great site! Phenomenal understanding of high level math and how to teach it.
calcworkshop.com has been a phenomenal experience! Difficult subject matter is explained clearly and I felt I learned it better than in class.
I decided to subscribe about 12 weeks into the 16 week Calculus I class. So far it has helped tremendously. I wish I had signed up at week 1. I look forward to learning with Jenn and the Calcworkshop for future classes.Thank you, Jenn!Regards,Frank
CalcWorkshop was a wonderful resource as someone going onto a college calculus 1 class after not having taken a math course in a year! Jenn always explained things in a logical order that built upon previous lessons, and the way she explained topics was always easy to understand. Calcworkshop saved this semester for me, and I couldn't be more grateful to have found this site. :)
Anyone struggling with math at any level would greatly benefit from this product. I signed up for it immediately after getting a D on my first exam in calculus 2 and ended up getting A's and B's on the remaining exams, getting a B overall in the class. Jenn breaks down the topics, without skipping any steps, in a way that makes them very easy to understand. I highly recommend this product!