| Euler's Number |
By definition, Euler's number (e) is the limit as n increases without bound of
(1+1/n)n, or the limit as v approaches 0 from the positive direction of (1+v)1/v, approximately 2.71828182846.
What is the limit, as n increases without bound, of (1+2/n)n? Derive your solution without the tools of Calculus. |
Problem Moderated by: Sasha |
| Problem Solution |
x = lim n -> ∞ (1+2/n)n = lim n -> ∞ (n+2/n)n
e = lim n -> ∞ (1+1/n)n = lim n -> ∞ (n+1/n)n
x/e = lim n -> ∞ (n+2/n+1)n.
Let u = n+1.
lim n -> ∞ (n+2/n+1)n = lim u -> ∞ (u+1/u)u-1
But as u increases without bound, u-1 is equivalent to u -- so,
lim u -> ∞ (u+1/u)u-1 = lim u-> ∞ (u+1/u)u = e
Therefore, x/e = e, so
x = e2 |
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