The Sum of a Geometric Series (Animated Proof)

What is the sum of a geometric series for a finite number of terms? What about an infinite number of terms? The video presents an animated proof of the famous formulas. Once you understand it, you can test yourself by solving the problem: what is 6 + 66 + 666 + .... + 66...6? The last term has 666 digits of 6.

Solution to 6 + 66 + 666 + .... + 66...6 = ?

If you like my videos, you can support me at Patreon:

Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you.

If you buy from the links below I may receive a commission for sales. This has no effect on the price for you.

My Books
"The Joy of Game Theory" shows how you can use math to out-think your competition. (rated 3.9/5 stars on 32 reviews)

"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.6/5 stars on 3 reviews)

"Math Puzzles Volume 1" features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 13 reviews.

"Math Puzzles Volume 2" is a sequel book with more great problems. (rated 4.3/5 stars on 4 reviews)

"Math Puzzles Volume 3" is the third in the series. (rated 3.8/5 stars on 5 reviews)

"40 Paradoxes in Logic, Probability, and Game Theory" contains thought-provoking and counter-intuitive results. (rated 4.3/5 stars on 12 reviews)

"The Best Mental Math Tricks" teaches how you can look like a math genius by solving problems in your head (rated 4.7/5 stars on 4 reviews)

"Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 5/5 stars on 3 reviews)

[iamyoda]

This is the first video in a long while *without* the catching phase "Can You Figure It Out". The transformation from math puzzles to math lessons has begun.

[SamvitAgarwal]

Hey. Although I do understand this proof itself and have seen many variations of it being used to prove other things about geometric series, I still do not understand the actual intuition behind multiplying by r. Sure, it allows us to cancel all the terms and arrive at a result, but what does multiplying by r actually represent?

In other words, my question is more or less if there is an intuitive way to stumble upon such a formula and how it might have actually been discovered in the first place.

[andymcl]

Hey Presh, I really like your videos. You have a nice way of explaining puzzles, and getting the workings of probabilities across. However, I feel this video doesn't really fit with the theme of the channel. You explained the concept well, but it's a relatively basic thing that many others have explained. If this is a new direction that you want to head in, fair enough, but I'm not sure I'll get enough out of it to continue watching, and I think a number of your fans might feel the same way. If it was just something you were trying out, that's fair enough too. After all, trial and error is a valid approach in maths!
All the best =]

[abtinshahidi]

There is also another series called the hypergeometric series(it uses the falling factorial and 4 parameters, only 1 of them being a variable.Its sum inf n = 0 (a)n * (b)n * z^n/ (c)n.Where a(n) is the pochammer symbol for falling factorial and |z| < 1.

[cikguhassan]

what software of app do you use to make all those smooth motion? very nice

[darshankrishnaswamy]

This is a great proof. Very simple and easy to understand with no complicated maths.

[shaharnuriel]

Hey Presh,
I wrote fairly simple riddle that might entertain your viewers:
Find a hole number that satisfy the following requirements
a: The unity digit is smaller than the ten's digit by 1
B: The hundred's digit is larger than the ten's digit by 2
C: The sum of the unity digit and the hundred's digit is 7
D: The digits sum is 10

[AnCoSt]

Great explanation! If you can remember the super-easy sum of an INFINITE geometric series (a/(1-r)), where a is the first term, and r is the common ratio, then you can always derive the sum for a FINITE geometric series of k terms by subtracting an INFINITE geometric series starting at the kth term from from an INFINITE geometric series starting at the first term. ie, a/(1-r) - ar^k/(1-r), which simplifies to a(1-r^k)/(1-r) .

edited out an exclamation point because it looked like a factorial...

[juliosantos]

Hi! It’s awesome!! What program do you use to do that amazing videos? Thanks!

[kapildeshmukh]

Our teacher used to call this technique as "Slide and subtract". Fun times!

[fumito]

Pretty fun derivation, did it today by finite induction as a homework exercise

[orestisg]

I would like to see videos like this more often

[jarencascino]

Always wondered about this. It was one of the few formulas my teacher said just to remember.

[BlinkLed]

I'd like to see more videos like this actually. This is a really easy example, but derivations for some more calculus oriented stuff would be really neat. Stuff like deriving power rule, or other calc identities would be really neat.

[someperson]

Of course, r <> 1 in the formula for S_k. Also,
-rS_k = -(r + r^2+ . . . + r^(k+1)). You left out the minus on the right hand side.

[redanwrong]

I like how he is trying to further explain something he talked about before, and everyone is just like "screw you, I don't want to learn math, how is this even related to your channel????"

[Uncle_Yam]

Simple to understand, but I haven't seen this derivation before

[GourangaPL]

somehow in all school books i've seen the ratio wasn't r, it was q... But i live in Poland, and the radio of geometric series is called "iloczyn" here so letter "i" is very problematic due to sums iterators and complex numbers, and "q" isn't used in polish language so it was free to use :)

[gthakur]

just for fun I ask ppl a problem of sum of infinite series 1+2+2^2 + and many gave me the answer as : -1. thats how i know whether they understand formula or not. they just mug up the formula for sum of infinite series is a/(1-r) but forget the applicable case. It's fun try with your friends .

[sunnykumar]

Any sum of series comes from method of difference ...multiplying something and subtracting it