Which of the following sequences are convergent and which are divergent?

a) $\displaystyle a_{n} = (1+2n)^{\frac{1}{n}}$

b) $\displaystyle a_{n} = cos(n\pi)$

c) $\displaystyle a_{n} = \frac{cos(3n)}{(1+n^{1/2})}$

This is what I got:

a) By the squeeze theorem, 1 < (1+2n)^1/2 < 3

and the sequence decreases as a(subscript n) > a(subscript n+1)

So is it correct to say that the sequence diverges?

b) {a(subscript n)} = {cosPI, cos2PI, cos3PI, cos4PI...}

= {-1,1,-1,1,...}

Since this sequence oscillates between -1 and 1, so it diverges.

Is this correct?

c) since cos(3n) <= 1 for all n, we have

cos(3n)/(1+n^1/2) <= 1/(1+n^1/2) <= 1/(n^1/2)

and we know that 1/n^1/2 is divergent (since p=1/2)

Is this correct?

I am almost sure that b and c are correct, but as for question a,

im lost... please help.. zee