So, i just took my exam. I think I did really well! Except, there were two questions that I sorta blanked out on and maybe was doing something wrong.

1) The first one was find the convergence of ((3n^2+8)/(8n^2+3))^n. I used the root test and found that the limit < 1 which means that the series converges absolutely.

However, the second part to this same series was whether the sequence ((3n^2+8)/(8n^2+3))^n converges or diverges? (if converges, compute lim as n-> inf of the seq).

---> so since the series converges, i was thinking that the sequence should converge (limit of seq. exists). However, when i tried to find the limit of the sequence, the sequence goes to inf (e^inf). Thus, i said that the sequence diverges to infinity.

My Question: shouldn't the sequence converge if the series converge? Why couldn't i find a limit for the sequence then?

2) Another question was simply write sin(5x)cos(5x) with maclaurin series using the trig identity : sin(2x) = 2sin(x)cos(x).

I knew the sin(5x) and cos(5x) series with the memorized maclaurin series. However, i was having trouble rewriting sin(5x)cos(5x)... I thought it would be

2sin(5x)cos(5x) = sin(10x) ---> sin(10x)/2.

I probably rewrote it wrong with the trig identity. I completely blanked on how to use these trig identity!