Prove that 1/2 + 1/4 + 1/8 + ... + 1/2^(n - 1) is less than 1
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- เผยแพร่เมื่อ 15 ต.ค. 2024
- In this video we prove that this geometric series is less than 1. We give a direct proof and not an induction proof. You could also do this with induction but I think the direct proof is more elegant. I hope this video is helpful:)
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Thank you:)
![Prove that 11^n - 4^n is divisible by 7 for any natural number, n. [Mathematical Induction]](http://i.ytimg.com/vi/pLbG3Vvpjjk/mqdefault.jpg)
![Prove that 11^n - 4^n is divisible by 7 for any natural number, n. [Mathematical Induction]](/img/tr.png)
Great trick, I thought of evaluating the sum as the geometric series though but this may be better
I’ve been reading through Stewart’s calculus (late transcendentals) and I found that I love infinite series and their manipulations 💚
awww me too! I love series!! I was at an old book store today and I got so many cool math books. I saw an older edition of Stewart's calculus there too:)
Isn't this Archemedes's Contradiction (one of them) that if you walk towards a wall, each step being half of the step of the last, then you will never reach the wall.
Yeah if you let the sum be infinite yes !!! Super cool❤️
What software are you using for these math videos??
Smoothdraw
How would you do this using induction?
Sir I have daubt , were take n= -1 so this formula false
"for all positive integers n>1"
Most people will see fractions, and make a run for it!!! LOL!!!
I forgot which mathematician added 1 to 100 super fast as a child and the teacher didn’t believe him. Anyhoo, this is almost the same thing but you are subtracting to simplify the equation. Then, matheMAGIC happens with fractions, with some logic of course.
Gauss. But the story is apocryphal.
Bravo.
Valla panikkum podo