Explained: "How do you solve this series question?∑1∞cos⁡(nπ)ln⁡(6n)"

Tristian Neal

Answered question

2022-04-10

How do you solve this series question?

\sum_{1}^{\infty} \frac{\cos (n π)}{\ln (6 n)}

Frain4i62

Beginner2022-04-11Added 16 answers

Since

\cos (n π) = {(- 1)}^{n}

, the series is

\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{\ln (6 n)}

which converges by the alternating series test because

\frac{1}{\ln (6 n)} \to 0

and

\frac{1}{\ln (6 n)}

is monotone decreasing.
But

\sum_{n = 1}^{\infty} \frac{1}{\ln (6 n)}

is not convergent because:
1. so is

\int_{1}^{\infty} \frac{d x}{\log (6 x)}

as can be seen by the limit test with

\int_{1}^{\infty} \frac{d x}{6 x}

applying L'Hôpital's rule.
2. Or by a direct comparison test with the divergent series

\sum_{n = 1}^{\infty} \frac{1}{6 n}

\sum_{n = 1}^{\infty} \frac{1}{\ln (6 n)} \geq \sum_{n = 1}^{\infty} \frac{1}{6 n}

because the harmonic series

\sum_{n = 1}^{\infty} \frac{1}{n}

is divergent.

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