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

Consider the sequence \{ f(n) \} defined by

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

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


We start by rearranging the expression for f(n),

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

Since each of the limits

    \[ \lim_{n \to \infty} \frac{1}{n+1} \qquad \text{and} \qquad \lim_{n \to \infty} \frac{1}{n} \]

exist and are equal to 0, we have that the sequence converges and

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

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