Not only does the sum diverge; the *sequence* diverges! Just look at the first two terms of the binomial expansion (all of whose terms are positive here, so the whole thing is greater than the first two terms). (1 + 1/n)^n² = 1 + n²/n + ... = 1 + n + ... and as n→∞, ... well, you get the picture.
I'm confused by this. As n approaches infinity for (1+1/n)^n, the inside of parenthesis will just be (1+0)^n, so it will be 1. How did it become 'e' ???!?
Why does Test for Divergence (nth term test) not work on this series? The inside part, 1 + 1/n, is always greater than 1, and the exponent n^2 is always greater than 1, therefore the whole term is greater than 1, and thus the limit is not 0. I believe using "the fact" the limit is actually e^n ≠ 0.
You are amazing at explaining and it's a joy to watch you solve these advanced problems with a smile on your face. Cheers to you friend!!
Nice video! Also digging the death star microphone.
Not only does the sum diverge; the *sequence* diverges!
Just look at the first two terms of the binomial expansion (all of whose terms are positive here, so the whole thing is greater than the first two terms).
(1 + 1/n)^n² = 1 + n²/n + ... = 1 + n + ...
and as n→∞, ... well, you get the picture.
I'm confused by this. As n approaches infinity for (1+1/n)^n, the inside of parenthesis will just be (1+0)^n, so it will be 1. How did it become 'e' ???!?
nvm working it out. I forgot 1^infinity is indeterminate and have to use L'hopital. watching 'the fact' video now.
How did it become e??
Because when u put limit n tends to infinity we get 1^infinity which is indeterminate form and u have to use natural log to solve it
What if it's raised to n^3 not n^2
Why does Test for Divergence (nth term test) not work on this series? The inside part, 1 + 1/n, is always greater than 1, and the exponent n^2 is always greater than 1, therefore the whole term is greater than 1, and thus the limit is not 0. I believe using "the fact" the limit is actually e^n ≠ 0.
Stephen Beck hmm did I say TFD doesn't work for this vid? I don't remember. In fact u could. The lim of a_n does go to inf thus the series div by TFD.
I believe you were referring to this video when you tried the root test on the other one and switched to TFD.
Using the L'Hopitals rule we can prove that.
lim n->infinity (1 + 1/n)^2 =e because of THE FACT!
You are wrong, instead of (1 + 1/n)^2 is :(1 + 1/n)^n
is because you can do natural log and manipulate the exponent down?
What to do if we get 1
What is the sum of √(1+((2r-1)/n)²) where n = infinity and r=1 to n
So anyone want to explain to me why this approaches e? Besides graphing it
Donno if its a stupid question but ,what is he holding in his hand?
a microphone
Because of the "fact" ;D
AndDiracisHisProphet yes!
Thank u sir
Thanks man
👏🏼🫶🏼