Absolute Convergence

• May 6th 2008, 09:40 PM
kithy
Absolute Convergence
I need help determining if these alternating series converge absolutely, converge conditionally or diverge. Give reasons.

a) the sum (from n=1 to infinity) of (-1)^n [tanh n / n ]
note tanh n>0 for all n>0

b) the sum (from n=1 to infinity) of (-1)^n [sinh(n / 8n+1)]
note: sinh n > 0 for all n > 0

Any help with these problems is greatly appreciated! (Hi)

K
• May 6th 2008, 10:48 PM
CaptainBlack
Quote:

Originally Posted by kithy
I need help determining if these alternating series converge absolutely, converge conditionally or diverge. Give reasons.

a) the sum (from n=1 to infinity) of (-1)^n [tanh n / n ]
note tanh n>0 for all n>0

For large $\displaystyle n$, $\displaystyle \tanh(n)$ goes to $\displaystyle 1$, so the summand behaves like $\displaystyle \frac{(-1)^n}{n}.$

So this is conditionaly convergent, but not absolutely convergent.

RonL
• May 6th 2008, 10:50 PM
CaptainBlack
Quote:

Originally Posted by kithy
I need help determining if these alternating series converge absolutely, converge conditionally or diverge. Give reasons.

b) the sum (from n=1 to infinity) of (-1)^n [sinh(n / 8n+1)]
note: sinh n > 0 for all n > 0

Here the summand behaves like $\displaystyle (-1)^n \sinh(1/8)$ for large $\displaystyle n$

RonL