This is really very confusingly written. Ithinkyou mean a_{n+1}where you wrote "a= n+1". That is, you are using the ratio test: which goes to 0< 1 as n goes to infinity so the series converges.

(But "a_{n+1}< a_{n}alone does NOT imply converges.)

As for what it converges to, [itex]\sum_{n=0}^\infty \frac{x^n}{n!}= e^x[/tex].

What??? sin(1/n)/(1/n) is certainly NOT equal to 1! And even if it were, that would NOT prove it was convergent. What are you trying to say here?b)a_{n}=nsin(1/n)

a_{n}=sin(1/n)/(1/n) =1 convergent

Yes, this is a geometric series with r> 1 and so diverges.a_{n}=(3/2)^{n}

3/2 >1 divergent