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/sci/ - Science & Math

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6200567 No.6200567[DELETED]  [Reply] [Original]

I found a way to approx e while being bored in calculus 1 class.

1.1^1=1.1
1.01^100=2.70
1.001^1000=2.716
1.000001^1000000=2.718280
1.000000001^1000000000=2.7182818270999…

e=2.718281828459045

It'll stay accurate up to the amount of digits after the decimal it seems. Has this already been discovered and can anyone give a proof if it has been? btw this is my first post on 4chan.

>> No.6200603

e is defined as
<div class="math">e = \lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n</div>
which is obviously equivalent to
<div class="math">e = \lim_{k \to \infty} \left(1 + \frac{1}{10^k} \right)^{10^k}</div>

>> No.6200608

>>6200603

rekt

>> No.6200612

Lol, how cute, OP.

>> No.6200609
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6200609

>>6200567
I mean shit nigger wat the fuck?

>> No.6200620

In particular (1+x/n)^n converges to e^x. You can find nice proofs of the convergence of (1+1/n)^n to e by googling "e as a limit". Look at this too: http://www.proofwiki.org/wiki/Equivalence_of_Exponential_Definitions

>> No.6200625
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6200625

Holy fuck

>> No.6200628

>>6200608
what do you mean, rekt? it's cool to independently rediscover these kind of things.

>> No.6200632

>>6200628

If you're OP, I honestly thought this was a kind of nice trick to find out about on your own. That said, >>6200603 explains why it's not particularly profound.

>> No.6200640

>>6200603
even tho it works since the limit is infinity, the limit should be evaluated for 10^k -> inf
otherwise i dont see how they are equivalent

>> No.6200642

>>6200640
You do not know that a subsequence of a convergent sequence converges to the same value?

>> No.6200647

>>6200640

The second expression represents what OP was doing. It also shows that substituting any expression for n in the original formula doesn't change the output.

Like >>6200642 said, using a larger term for n is basically just "starting" the series summation from a "checkpoint" if you will.

>> No.6200767

>>6200567
>Has this already been discovered
Look, it's great that you're figuring things out on your own, but don't start thinking you're some kind of beautiful mind