Calc series help - converge absolutely conditionally or diverge?

Question

Can someone help me out with these review questions? Who gives a test the day before Thanksgiving?! Argh..haha..thanks!!

Show if these series Converge conditionally, converge absolutely or diverge.
(all summation from 1 to infinity)

1. ((-1)^k) / ((k(k+1))^(1/2))

2. ((-1)^(k)(k!)^(2)) / ((2k)!)

3. ((-1)^(k)k!) / (2^(k))

I know there are three options and three questions but it doesn't mean that one is assigned to each, for example they could all converge absolutely. I just have no clue how to prove this and my prof canceled office hours today because it's snowinggggg =O

Answer 1

1) in abs value is of the same class as 1/n divergent
lim I a_n I=0
Ia_n+1I take the function y=1/sqrt(x^2+x)
y´= 1(/x^2+1)* 1/2 sqrt(x^+x) *(-2x-1)<0 Ia_nI is decreasing
There is a criteria which assures that this series is conditionally convergent
2) take abs val.and use ratio test
Ia_n+1I/I a_nI = (n+1)^2/(2n+1)*(2n+2)==>1/4 <1 absolutely convergent
3)The same
Ia_n+1I/Ia_nI = (k+1)/2==> infinity
so Ia_nI is increasing and its limit is not zero .So lim an is not zero and the series is divergent

Answer 2

i could suggest alternating series test on the first one... it shows convergence.. to determine whether absolute or conditional... use comparison test...

for the next ... you could use ratio test... since the expression involves factorials...


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