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Determine the convergence of the series (-1)n / (log (en + e-n))

Consider the series

    \[ \sum_{n=1}^{\infty} \frac{(-1)^n}{\log(e^n + e^{-n})}. \]

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.


The given series converges conditionally.

Proof. First, we have

    \[ \lim_{n \to \infty} \frac{1}{\log(e^n + e^{-n})} < \lim_{n \to \infty} \frac{1}{\log (e^n)} = \lim_{n \to \infty} \frac{1}{n} = 0. \]

Furthermore, the sequence of terms a_n = \frac{1}{\log(e^n + e^{-n})} is monotonically decreasing. We can see this by looking at

    \[ f(x) = \frac{1}{\log(e^x+ e^{-x})} \quad \implies \quad f'(x) = \frac{e^{-x} - e^x}{(e^x+e^{-x})(\log (e^x + e^{-x}))^2}. \]

This derivative is negative for all x > 0 (since the numerator is negative and the denominator is always positive). Hence, the sequence \{ a_n \} is monotonically decreasing. Therefore, by the Leibniz rule we have

    \[ \sum_{n=1}^{\infty} \frac{(-1)^n}{\log(e^n + e^{-n})} \]

converges.

This convergence is conditional since

    \[ \sum_{n=1}^{\infty} \left| \frac{(-1)^n}{\log(e^n + e^{-n})} \right| = \sum_{n=1}^{\infty} \frac{1}{\log(e^n + e^{-n})}. \]

This series diverges by the comparison test since

    \[ \frac{1}{\log (e^n + e^{-n})} \geq \frac{1}{\log(2e^n)} = \frac{1}{n + \log 2} \geq \frac{1}{2n}, \]

and

    \[ \sum_{n=1}^{\infty}\frac{1}{2n} = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n} \]

diverges. \qquad \blacksquare

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