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Determine the convergence or divergence of f(n) = (n2 + 3n – 2) / (5n2)

Consider the function f(n) defined by

    \[ f(n) = \frac{n^2 + 3n - 2}{5n^2}. \]

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


Simplifying f(n) we have

    \begin{align*}  f(n) &= \frac{n^2 + 3n - 2}{5n^2} \\[9pt]  &= \frac{1 + \frac{3}{n} - \frac{2}{n^2}}{5}. \end{align*}

Therefore,

    \[ \lim_{n \to \infty} f(n) = \frac{1}{5} \cdot \lim_{n \to \infty} \left( 1 + \frac{3}{n} - \frac{2}{n^2} \right) = \frac{1}{5}. \]

Hence, the sequence \{ f(n) \} converges and has limits \frac{1}{5}.

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