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Determine the convergence or divergence of f(n) = 100,000n / (1 + n2)

Consider the function f(n) defined by

    \[ f(n) = \frac{100000n}{1+n^2}. \]

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


We have

    \[ f(n) = \frac{100000n}{1+n^2} = 100000 \left( \frac{n}{1+n^2} \right). \]

Then,

    \[ \lim_{n \to \infty} f(n) = 100000 \cdot \lim_{n \to \infty} \frac{\frac{1}{n}}{\frac{1}{n^2} + 1} = 0. \]

Hence \{ f(n) \} converges to 0.

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