# Convergent/Divergent test: $\sum\limits_{n = 1}^\infty e^{2n} \cos(\pi n)$

Trying to figure out if I use the alternating series test or divergence test.

$$\sum_{n = 1}^\infty e^{2n} \cos(\pi n)$$

I believe $\cos$ will be between $-1$ and $1$ and while $$n \to \infty$$ $$e^{2n} \to \infty$$ Does this mean it is divergent and I should use test for divergence ?

• Do the terms go to $0$? – lulu Jun 15 '17 at 13:15

The simplest test or first check, based on the negation of: $$\sum a_n \; \mbox{converges} \implies a_n \to 0$$ yields (for any series, not necessarily alternating): $$\lim_{n \to \infty} a_n \ne 0 \implies \sum a_n \; \mbox{diverges}$$ Now note that $e^{2n} \cos(\pi n) \not\to 0$.