Sequences converge or diverge?

[zeion]

Homework Statement

I need to see if these sequences converge or diverge:

1) $$a_n = ncosn\pi$$


2) $${ 0,1,0,0,1,0,0,0,1,0,0,0,0,1,... }$$

3) $$a_n = \frac{1 . 3 . 5 . ... (2n - 1)}{n!}$$

Homework Equations

The Attempt at a Solution

1) [tex]]cosn\pi = -1[/itex] so [itex]a_n \to -\infty[/tex]?

2) What is this. where is the n?

3) Not sure

 

[Matterwave]

1) are you sure cos(n*pi)=-1 always? What's cos(0) or cos(2*pi)?

2) There is no n, it's a pattern sequence.

3) What convergence/divergence tests have you thought of applying to this? Have you gotten anywhere with it? The last course I took that involved proving convergence/divergence was 3 years ago...so I don't remember every test that could be applied.

 

[Dick]

I wouldn't consider an->(-infinity) to be convergent. For 2) it appears to be a sequence with an infinite number of 0's and 1's in it. I wouldn't be distracted by trying to figure out a pattern. Who know's what in the '...'? It could be '0,0,0,0,0,0,0,0,0,0,0' forever. Just assume alternating 1's and 0's continue infinitely. What about convergence? For 3) you can apply a ratio test to a sequence. Each additional factor in the numerator is roughly twice the size of the corresponding factor the denominator. What's you gut feeling?