Sum to Infinity of a Geometric Series

[odolwa99]

Homework Statement

Q.: A geometric series has first term 1 and common ratio \frac{1}{2}sin2\theta. Find the sum of the first 10 terms when \theta = \frac{\pi}{4}, giving your answer in the form h - \frac{1}{2^k}, where h, k \in N.

Homework Equations

Sn = \frac{a(1 - r^n)}{1 - r}, when IrI < 1

The Attempt at a Solution

S10 = \frac{a(1 - r^n)}{1 - r}

S10 = \frac{1(1 - (1/2)sin2(\pi/ 4) to the power of 10}{1 - (1/2)sin2(\pi/4)} ... The <sup>tags did not take, for some reason, hence the 'to the power of 10'. Sorry about that.

I'm stuck here because if I solve for \frac{1}{2}sin2\theta, I get a decimal answer = 0.013706. The question wants the answer defined as whole numbers but I'm unable to work out the next step in the sequence. Can someone help and give me a tip on what to do next? Thanks.

Ans.: From textbook: 2 - (1/ 29)</sup>

 

[micromass]

I'm stuck here because if I solve for \frac{1}{2}sin2\theta, I get a decimal answer = 0.013706.

How did you get that?

\sin(2\theta)=\sin(\pi/2)

has a nice answer, no?

 

[odolwa99]

Ok, so the sum is now...

\frac{1 - (1/2)sin(\pi/ 2)tothepowerof10}{1 - (1/2)sin(\pi/ 2)}

so...

1 - (1/2)sin(\pi/ 2)¹⁰ = 1 - (1/2)sin(\pi/ 2)
1 - 1 - (1/2)sin(\pi/ 2)¹⁰ + (1/2)sin(\pi/ 2)
- (1/2)sin(\pi/ 2)⁹

Ok, obviously I'm wrong but this is what my attempt looks like. Thanks for the help so far but can you help me out a little bit more...please? Thanks.

 

[micromass]

What is \sin(\pi /2)?? Isn't that a nice value?

 

[odolwa99]

Err...In a right angled triangle, Sin( \theta ) = Opposite/ Hypotenuse...?

 

[micromass]

Err...In a right angled triangle, Sin( \theta ) = Opposite/ Hypotenuse...?

Go check your trigonometric formula's...

You should really know things like \sin(0) and \sin(\pi/2) by heart.

 

[odolwa99]

Ok. Sorry about all this. It's literally been years since I last studied maths and I'm having to begin from scratch all over again. You'll have to be patient with this old man. I'll see what I can find out from your advice and show my work tomorrow. Good night.

 

[micromass]

Ok. Sorry about all this. It's literally been years since I last studied maths and I'm having to begin from scratch all over again. You'll have to be patient with this old man. I'll see what I can find out from your advice and show my work tomorrow. Good night.

It's alright. This might be helpful: http://en.wikipedia.org/wiki/Trigonometric_identities

 

[odolwa99]

Ok, so here is, hopefully, my final attempt...

S10 = \frac{1(1 - (1/2)sin2(\pi/ 4)tothepowerof10)}{1 - (1/2)sin2(\pi/ 4)}

S10 = \frac{1 - (1/2)sin(2\pi/ 4)tothepowerof10}{1 - (1/2)sin(2\pi/ 4)}

S10 = \frac{1 - (1/2)sin(\pi/ 2)tothepowerof10}{1 - (1/2)sin(\pi/ 2)}

S10 = \frac{1 - (1/2)(\sqrt{4}/2)tothepowerof10}{1 - (1/2)(\sqrt{4}/2)}

S10 = \frac{1 - (2/ 4)tothepowerof10}{1 - (2/ 4)}

S10 = \frac{1 - (1/2)tothepowerof10}{(1/2)}

S10 = 2 - \frac{1}{2^9}

 

[HallsofIvy]

Are you talking about sin^{10}(\pi/2) or sin((\pi/2)^{10})?

sin(\pi/2)= 1, but why in the world would you write that as "\sqrt{4}/2"?

 

[Mark44]

Are you talking about sin^{10}(\pi/2) or sin((\pi/2)^{10})?

sin(\pi/2)= 1, but why in the world would you write that as "\sqrt{4}/2"?

I've seen some texts present the trig functions in this way as a mnemnonic device.

sin(0)= \sqrt{0}/2
sin(\pi/6)= \sqrt{1}/2
sin(\pi/4)= \sqrt{2}/2
sin(\pi/3)= \sqrt{3}/2
sin(\pi/2)= \sqrt{4}/2

The OP might have been referring to something like this, and didn't realize that \sqrt{4}/2 could and should be replaced with 1.

 

[odolwa99]

Based on the answer, I solved the question as sin((\pi/ 2)¹⁰).

As regards, expressing sin(\pi/ 4) as \frac{\sqrt{4}}{2}; yes, I was using the mnemonic but, as I indicated above, this is the first time I've dealt with these kinds of equations so I'll express it as 1, henceforth.

Thanks for helping, guys. I really do appreciate it.