10 - Series and Sigma Summation Notation - Part 1 (Geometric Series & Infinite Series)

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In this lesson, you will learn what a series is in math and how they compare to sequences. A series is the sum of the terms of a sequence. We have already discussed some special sequences such as the arithmetic sequence and geometric sequence. We now extend this idea and discuss the arithmetic and geometric series, infinite series, finite series, and more. We also spend a great deal of time introducing the sigma notation for summation, which is how we actually write the series down when solving problems..

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[@rrr00bb1]

whenever possible, it's helpful to define infinite series using recursion. you can just solve for its value in most cases: S = 1 + (1/2)S. (1-(1/2))S = 1. (1/2)S = 1. S=2. S= (1/2)^0 + (1/2)^1 + (1/2)^2 + ... + (1/2)^n + (1/2)^(n+1) S. = 1 + 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + (1/2)^(n+1)S. Note that the last term is multiplied times S. It's the tail of S expanded n times. But something AMAZING happens if you are careful. S isn't really the value. If the tail of the recursion goes to zero S = Sum(S,n), when n goes to infinity, because Tail(S,n) goes to zero. S = Sum(S,n) + Tail(S,n). This lets you calculate the closed form formula for n terms! S - Tail(S,n) = Sum(S,n). S - (1/2)^(n+1)S = Sum(S,n). S(1 - (1/2)^(n+1)) = Sum(S,n). 2(1 - (1/2)^(n+1)) = Sum(S,n). 2 - (1/2)^n = Sum(S,n). try it out... adding the first 4 terms 0..3 replacements is... Sum(S,3) = 1/1 + 1/2 + 1/4 +1/8 = 15/8 = 2-(1/2)^3 = 2 - (1/8) = (2*8-1)/8 = 15/8 I honestly have no idea how people figure out most infinite series closed forms without using recursion. Note the sum like 1/2 + 1/4 + 1/8 + ... = 1: S = 1/2 + 1/2 S. 2S = 1 + S. S = 1. S = 1/2 + 1/2 (1/2 + 1/2 S) = 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + 1/(2^(n+1))S. and this to top it off: S = 0.9 + 0.1 S 10 S = 9 + S 9 S = 9 S = 1 S = 1 = 0.9 + 0.1(0.9 + 0.1 S) = 0.9 + 0.09 + 0.01(0.9 + 0.1 S) = 0.9999.... And this works perfectly well for divergent series like "-1 = 1+2+4+8+...", where it is very clear what's going on. S=Sum(S,n) only when Tail(S,n) is zero; S is an important number in calculating the closed form; and is not necessarily the total. It is part of the recursive definition that replaces infinite iteration. -1/12 = S -1 = 12 S 1 = -12 S 1 = (1-13)S 13 S + 1 = S S = 1 + 13 S = 13^0 + 13 S = 13^0 + 13(13^0 + 13 S) = 13^0 + 13^1 + 13^2 + ... + 13^n + 13^(n+1) S subtract the tail, and you have a formula for the first 13 powers. more generally... A = 1 + x A (1-x)A = 1/(1-x) When you differentiate this, you get the famous "-1/12 = 1+2+3+4+..." strange sequence. It is no paradox though because S isn't the value. S-Tail(S,n) is the value, and that goes to infinity as n goes to infinity, and it gives you the n(n+1)/2 formula, in case you didn't know it.

[@mrinfinity1217]

Thank you so much...i have exam this Monday and our teacher told us nothing about sigma, but this though me everything

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[@LarryZAR]

9:11 In order for you to get to 1 you have to get to 1/2 before that 1/4 before that 1/8 etc Therefore ...... sum of (1/2+1/4+1/8...)=1

[@MHatch2008]

This was awesome! I'm having trouble finding Part 2 to see the other direction of going from series to summation notation?

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Great lesson👍. Bonus sad lesson included. I should have verified parts two and three being posted before watching this. Now i have to start over somewhere else.

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[@phoetumaung2523]

The sum of the first 5 terms of a G.P is 4 and the sum of the terms from the fourth to the eighth inclusive is 125/16. Find the common ratio and the sixth term.

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It would be great if you had links or names of the previous lessons you mentioned!

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So impressive 🤞 11:08

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Accuracy is important than perfection. Eventhough we are perfect in many aspects,we are not quite sure of its accuracy.

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im really waiting for part 2 ... pls do part 2 pls

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[@LarryZAR]

9:11 listen to ted ed (zenos paradox) then keep on watching.

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I've always wondered what those sigmas were all about. Now I finally know.

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[@DSCuber-28-01-2019]

I couldn't find the video where you show how to calculate the series. Will you upload it in the future?

[@VoidAshen]

here use it in a header file in c++ or make a void function in c#, java or whatever int Sum(int iniVal, int count, int step) { int result = 0; for (int n = iniVal; n

[@iloverihanna]

Thank you sooo much

[@razivgreenway8041]

Where can I find part 2 of this video?

[@pichytechno6782]

Nice video truly but I have a doubt why should we call it finite series in arithmetic series when we can also add an infinite amount of numbers same as in infinite series?

[@heatbeat445]

You saidn times any term Now you're squaring n How both of them can you the same answer ... Means 2 times n or 2×n wil give you 100 if n= 50 But in the second explanation how n^2 gives you 20 if n=10 Bcz 10^2 =100

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Please explain fractional binomial

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Sir i do not understand the should be the base of a number is that right ?

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[@MathAndScience]

Actually what you refer to is called “special relativity”. General relativity is the theory of gravity. Time flows different in a gravitational field!

[@rasitalpatasoy3866]

I couldn't find the next video. Can someone help?

[@zyreljohn3082]

What if the expression's has negative and positive, example like 1-2+3-4+5

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Thanku very much!

[@brewfruit]

Ok, I was with you until you got to 28:41…. How does 2 to the 3rd power give you 8 and etc? I got lost there.

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2 times 2 is 4 and times 2 is 8

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What category is this under on your site. I looked for it but couldn't find it..

[@MathAndScience]

It’s not on the site yet. Just released this lesson for free on KZbin. I’m working on the rest along with worksheets and quizzes to go with these lessons. There are many more forthcoming for members!

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@@MathAndScience yes.. i was a member before.. I was too busy with work and cancelled but now I am ready to join again. I was wondering if this was pre-calculus or Algebra.. Thanks for getting back to me.

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where can i find the next part?

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Is the increment for k always 1?

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Wer is part 3..?

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16:24

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Where is part 2?

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part two

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What the sigma

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Ofogog

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[@josephciaravino4115]

This man is a gift to humanity! 😆 I love how he takes time to point things out any sensible person would just accept as given and breeze over. 👍Thank you!

[@ULTIMARAGNAR0K]

Where is part 2?

[@laizaabigailinocencio7194]

25:41

[@josephciaravino4115]

This man is a gift to humanity! 😆 I love how he takes time to point things out any sensible person would just accept as given and breeze over. 👍Thank you!

[@NolstanLogic]

Where is the part 2