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微分應用題連續複利

Oct 9th 2013, 14:49

http://zh.wikipedia.org/wiki/E_(数学常数)

請參考第一個定義

e = lim_{n to inf}(1 + 1/n)^n

那麼先看(1+0.05/n)^n*15，我們先試著整理( )的內部，變成 1 + 1/t的樣子

令 0.05/n = 1/t
=> n = 0.05t

所以 (1+0.05/n)^n*15 = (1 + 1/t)^(0.05t * 15)
當 n to inf => t to inf
所以加上lim會變成 lim_{t to inf} (1 + 1/t)^(0.05t * 15)
= lim_{t to inf} ((1 + 1/t)^t)^(0.05 * 15) [ ps a^(xy) = (a^x)^y ]
= e^(0.05 * 15)

參考資料維基

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