absolutely convergent, conditionally convergent or divergent

**acosta0809** · August 4th 2009, 12:39 PM

the question is: determine whether the series is absolutely convergent, conditionally convergent or divergent.

$\sum_{n = 1}^{\infty} (-3)^n \frac{n+1}{\exp(n)}$

I found that if i do the ratio test it diverges but im not sure how to do the alternating test, is a just

$\frac{n+1}{\exp(n)}$

or does it include the 3?

**skeeter** · August 4th 2009, 01:05 PM

Originally Posted by acosta0809

the question is: determine whether the series is absolutely convergent, conditionally convergent or divergent.

$\sum_{n = 1}^{\infty} (-3)^n \frac{n+1}{\exp(n)}$

I found that if i do the ratio test it diverges but im not sure how to do the alternating test, is a just

$\frac{n+1}{\exp(n)}$

or does it include the 3?

ratio test ...

$\lim_{n \to \infty} \left| \frac{(-3)^{n+1}(n+2)}{e^{n+1}} \cdot \frac{e^n}{(-3)^n (n+1)}\right|$

$\lim_{n \to \infty} \left| \frac{(-3)(n+2)}{e(n+1)}\right|$

$\left|\frac{-3}{e}\right| \lim_{n \to \infty} \frac{n+2}{n+1}$

$\left|\frac{-3}{e}\right| \cdot 1 > 1$

series diverges

**Krizalid** · August 4th 2009, 01:05 PM

For each $n\ge1$ it's $\frac{3^{n}(n+1)}{e^{n}}>n+1,$ so this is far to converge absolutely.

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