How to solve this? The first three terms of a different geometric progression are (4/3)cos^2(3theta), (16/9)cos^4(3theta), and (64/27)cos^6(3theta)

[Kulla_9289]

r = 4(cos^2(3theta))/3. I found the angles as pi/18, 11pi/18, 5pi/18, 7pi/18. I am stuck here.

 

[blamocur]

r = 4(cos^2(3theta))/3.

Looks good.

I found the angles as pi/18, 11pi/18, 5pi/18, 7pi/18. I am stuck here.

Doesn't look good. For which values of $r$ will the sum of the geometric progression diverge?

 

[Kulla_9289]

1<r<-1?

 

[blamocur]

1<r<-1?

Right.
Can you now translate this to a condition on $\theta$ ?

 

[Kulla_9289]

pi/18 < r < 5pi/18

 

[Kulla_9289]

the solution given is this: I dont understand how they got lesser than or equal to sign and theta

 

[blamocur]

Right

Not quite actually. The condition $-1<r<1$ is the condition for the convergence of the sum. But the sum can diverge without its limit being infinity.
Can you see that?

 

[Kulla_9289]

I don't get what converge and diverge means

 

[blamocur]

I don't get what converge and diverge means

What are you studying? Are you familiar with limits? You might find this Wikipedia page helpful, but feel free to get back to us if you have questions.

 

[Kulla_9289]

I am studying those but i just memorised -1 <r<1 without really understanding

 

[blamocur]

I am studying those but i just memorised -1 <r<1 without really understanding

Convergence means that partial sums have a (finite) limit. Divergent series do not have limit, including but not limited to the case where they grow to infinity. Take a look at partial sums in the case $r=-1$.
Also, what do you think of the Wikipedia page I've mentioned earlier ?

 

[Kulla_9289]

I got that it's a divergent series. Now what?

 

[blamocur]

I got that it's a divergent series. Now what?

What do you mean?
For which $r$'s your sum diverges to infinity?

 

[Kulla_9289]

|r|>=1
So pi/18 and 5pi/18.

 

[Kulla_9289]

how do i determine whether a series is converging or diverging?

 

[blamocur]

|r|>=1

What are the partial sums for $r=-1$?

how do i determine whether a series is converging or diverging?

Look at the partial sums and see if you can figure out their limits.

 

[Kulla_9289]

limits

What do you mean? Do you mean 0<=y<=4/3?

 

[Kulla_9289]

Look at the partial sums and see if you can figure out their limits.

so 4(cos^2(3theta))/3, substituting pi/18 and 5pi/18 consecutively gets me to 1, hence the series is a diverging one.

 

[Kulla_9289]

But how do i write it with inequality signs?

 

[blamocur]

But how do i write it with inequality signs?

First figure out inequalities for $r$ in the general case. Can you write an expression for partial sums when $r=\pm 1$?
Once you get this figured out you will likely see the answer to your post #6.