1. ## Converging/ Diverging Series

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑(n=1,∞) (-1)^n/n√n

∑(n=1,∞) [(-1)^(n-1)][n/n(^2)+1]

My textbook has pretty sparse information in this chapter and I'm really having some trouble understanding the examples. What is the difference between series that converge conditionally and absolutely?

2. Originally Posted by schnek
Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑(n=1,∞) (-1)^n/n√n

∑(n=1,∞) [(-1)^(n-1)][n/n(^2)+1]

My textbook has pretty sparse information in this chapter and I'm really having some trouble understanding the examples. What is the difference between series that converge conditionally and absolutely?
$\sum_{n=1}^{\infty}\frac{(-1)^n}{n\sqrt{n}} = \sum_{n=1}^{\infty}\frac{(-1)^n}{n^{3/2}}$ Taking the absolute value of the summand gives you a $p$-series with $p=3/2$. What does this tell you?

$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{n^2+1}$ Use the alternating series test. As for whether it converges absolutely, compare it to $\sum_{n=1}^{\infty}\frac{1}{n}$.

3. Originally Posted by schnek
Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑(n=1,∞) (-1)^n/n√n

∑(n=1,∞) [(-1)^(n-1)][n/n(^2)+1]

My textbook has pretty sparse information in this chapter and I'm really having some trouble understanding the examples. What is the difference between series that converge conditionally and absolutely?
Let $\sum_{n=1}^{\infty}a_{n}$ represent an arbitrary infinite series. If $\sum_{n=1}^{\infty}|a_{n}|$ converges, then the orginal series $\sum_{n=1}^{\infty}a_{n}$ converges absolutely. However, if the series $\sum_{n=1}^{\infty}a_{n}$ converges, but $\sum_{n=1}^{\infty}|a_{n}|$ does not converges, then the orginal series $\sum_{n=1}^{\infty}a_{n}$ converges conditionally.

I was too slow! Hope it helps anyways.