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Determine the convergence or divergence of f(n) = (n2 / (n+1)) – ((n2 + 1) / n)

by RoRi

Consider the sequence f(n) defined by

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

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

First, we rewrite f(n) as follows,

    \begin{align*} f(n) &= \frac{n^2}{n+1} - \frac{n^2+1}{n} \\[9pt] &= \frac{n^3 - n^3 - n^2 - n - 1}{n^2+n} \\[9pt] &= - \left( \frac{1}{1 + \frac{1}{n}} + \frac{1}{n+1} + \frac{1}{n^2+n} \right) \\[9pt] &= - \left( \frac{1}{1+\frac{1}{n}} + \frac{\frac{1}{n}}{1 + \frac{1}{n}} + \frac{ \frac{1}{n^2}}{1+ \frac{1}{n}} \right). \end{align*}

Since each of these limits exists we have

    \[ \lim_{n \to \infty} f(n) = - \left( 1 + 0 + 0 \right) = -1. \]

Hence, the sequence \{ f_n \} is convergent with limit -1.

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