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

Consider the function f(n) defined by

    \[ f(n) = \frac{n}{2^n}. \]

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


We consider the function f(x) = \frac{x}{2^x}. From our work in Chapter 7 (either by Theorem 7.11 or by L’Hopital’s rule) we know

    \[ \lim_{x \to \infty} \frac{x}{2^x} = 0. \]

Hence,

    \[ \lim_{n \to \infty} f(n) = 0. \]

So, the sequence \{ f(n) \} converges with limit 0.

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