Determine the convergence of the series (-1)n ((2n+100) / (3n+1))n

Consider the series

$\sum_{n=1}^{\infty} (-1)^n \left( \frac{2n+100}{3n+1} \right)^n.$

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

The given series is absolutely convergent.

Proof. To see this is absolutely convergent, first we have

$\sum_{n=1}^{\infty} \left| (-1)^n \left( \frac{2n+100}{3n+1} \right)^n \right| = \sum_{n=1}^{\infty} \left( \frac{2n+100}{3n+1} \right)^n.$

Then, looking to apply the root test, we have

$a_n = \left( \frac{2n+100}{3n+1} \right)^n \quad \implies \quad a_n^{\frac{1}{n}} = \frac{2n+100}{3n+1}.$

And so,

$\lim_{n \to \infty} \frac{2n+100}{3n+1} = \frac{2}{3} < 1.$

Hence, by the root test, the series converges. Therefore, the given series is absolutely convergent $. \qquad \blacksquare$

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