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Determine the convergence or divergence of f(n) = (1 + 2/n)n

Consider the function f(n) defined by

    \[ f(n) = \left( 1 + \frac{2}{n} \right)^n. \]

Determine whether the sequence \{ f(n) \} converges or diverges, and if it converges find the limit.


By property (10.13) on page 380 of Apostol we know

    \[ \lim_{n \to \infty} \left( 1 + \frac{a}{n} \right)^n = e^a. \]

So, in the present case we have

    \[ \lim_{n \to \infty} f(n) = \lim_{n \to \infty} \left( 1 + \frac{2}{n} \right)^n = e^2. \]

Therefore, \{ f_n \} converges to the limit e^2.

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