| The Power of e |
By definition, Euler's number (e) is the limit, as n increases without bound, of
(1+1/n)n, approximately 2.71828182846. What is the limit, as n increases without bound, of
(1+2/n)n? |
Problem Moderated by: Douglas |
| 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 = e
We know this to be true on the basis that no further operations in terms of u are performed upon the exponent, which allows u and u-1 and u+1,000,000 to be equivalent values as u increases without bound.
Therefore, x/e = e, so
x = e2 |
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