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Determine the convergence or divergence of f(n) = (3n + (-2)n) / (3n+1 + (-2)n+1)

by RoRi

Consider the function f(n) defined by

    \[ f(n) = \frac{3^n + (-2)^n}{3^{n+1} + (-2)^{n+1}}. \]

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

We divide the numerator and denominator of the given expression by 3^n,

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

Since

    \[ \lim_{n \to \infty} 1 + \left(-\frac{2}{3}\right)^n = 1 \quad \text{and} \quad \lim_{n \to \infty} 3 - 2 \left(-\frac{2}{3}\right)^n = 3 \]

we have

    \[ \lim_{n \to \infty} f(n) = \frac{1}{3}. \]

So, \{ f_n \} converges to the limit \frac{1}{3}.

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