1. ## Converging Sequences

Ok.I cant think how to start either of these problems.

For each sequence, determine whether the sequence converges and if it does, find the limit.

a) (e^(npii))/1+3^n

b) sinh[(npii)/2]

a) I presuming that the dominant term is e, but when I divide by it I get

1/[(1/e^npii)+(3^n)/e^npii]

Is this the right way of doing it? If not can someone point me in the right direction.

sinh[(npii)/2]=(e^(npii/2) - e^(-npii/2))

I tired a way using the the triangle inequality followed by the squeeze rule, but that came out wrong, so I dont know where I went wrong.

Thanx

Bex

2. Originally Posted by bex23
Ok.I cant think how to start either of these problems.

For each sequence, determine whether the sequence converges and if it does, find the limit.

a) (e^(npii))/1+3^n
Can we assume that this is (e^(npi i)}/(1+ 3^n)?
Is so, that's east: |e^(npi i)|<= 1 while the denominator increases without limit.

b) sinh[(npii)/2]
sinh(z)= (e^(z)- e^(-z))/2i so sinh(npi i)/2= [(e^(n pi i)- e^(n pi i))/2] (1/2i)= sin(n pi) (-i/2) and sin(n pi)= 0 for all n.

a) I presuming that the dominant term is e, but when I divide by it I get

1/[(1/e^npii)+(3^n)/e^npii]

Is this the right way of doing it? If not can someone point me in the right direction.

sinh[(npii)/2]=(e^(npii/2) - e^(-npii/2))

I tired a way using the the triangle inequality followed by the squeeze rule, but that came out wrong, so I dont know where I went wrong.

Thanx

Bex

3. Originally Posted by HallsofIvy
Can we assume that this is (e^(npi i)}/(1+ 3^n)?
Is so, that's east: |e^(npi i)|<= 1 while the denominator increases without limit..
So this means the sequence is divergent right?

Originally Posted by HallsofIvy
sinh(z)= (e^(z)- e^(-z))/2i so sinh(npi i)/2= [(e^(n pi i)- e^(n pi i))/2] (1/2i)= sin(n pi) (-i/2) and sin(n pi)= 0 for all n.
Not sure that this is correct. But isnt sinh(z)= (e^(z)- e^(-z))/2

The formula I have does not include i. That would be sin(z) not sinh(z). Also the values seem to have got mixed up. It should be

sinh[(npii)/2]

Hopefully you can help me out.

Thanx

4. Originally Posted by bex23
So this means the sequence is divergent right?
It means it's convergent, it converges to 0.

5. Originally Posted by Showcase_22
It means it's convergent, it converges to 0.
But isnt

e^(pii)=-1 so

e^(npii)=(e^pii)^n= (-1)^n

and so is divergent

6. Factor in the denominator and it converges to 0.....

Am I looking at the right sequence?

7. Originally Posted by Showcase_22
Factor in the denominator and it converges to 0.....
Can you maybe explain what you mean. I am having a slow day today, lol

Originally Posted by Showcase_22
Am I looking at the right sequence?
The sequence is the part (a) one. That is

[e^(npii)]/(1+3^n)

8. Isn't $\displaystyle a_n=\frac{(-1)^n}{1+3^n}$?