Which sequence converges?

Question

Which sequence converges?

 

a) an=n+3/n

 

b) an=-1+(-1)^n/n

 

c) an=sin (n*pi)/2

 

d) an=n!/3^n

 

e) an=n/(ln(n))

 

I know that I have to take the limit as n approaches infinity and see if it converges, but I'm not good at taking limits. Please show all your work. Thank you. The n after an is lower case. 

Answer 1

a) an=n+3/n

 

As n increase, n increases, ;-) , and 3/n decreases (3/2 is much bigger than 3/100).

 

So an increases and the sequence does not converge.

 

b) an=-1+(-1)^n/n

 

You can use the Squeeze Theorem, I think:

 

-1+ (-1)/n ≤ -1+(-1)^n/n ≤ -1+1/n

 

Since the left and right sides go to -1, then the center does too.

 

c) an=sin (n*pi/2)

 

This does not converge because it just oscillates between ±1.

 

d) an=n!/3^n

 

After n = 7, this is a constantly increasing function (graph it); so its sequence does not converge.

 

e) an=n/(ln(n))

 

ln(n) increases much more slowly than n, so an increases and its sequence does not converge.