Series involving trig functions

[Noober]

Homework Statement

http://img87.imageshack.us/img87/6784/dsc07215xt5.jpg
http://img72.imageshack.us/img72/9435/dsc07216aa0.jpg

Homework Equations

None.

The Attempt at a Solution

Answers for part I are circled, here's parts II-IV.
http://img485.imageshack.us/img485/8642/dsc07217uq2.jpg
http://img485.imageshack.us/img485/6999/dsc07218qf2.jpg

Any help would be appreciated. This is a very important homework, as you can tell by the title.

If the pics are too small, download them and zoom in, or use the FF extension ImageZoom.

 

[Gib Z]

2) Derive a taylor series centered at x= pi/3 and just keep summing terms. You will know you have 4 places accuracy when after summing a certain number of terms for 4 places are still the same, or also when the terms become smaller than 0.0001.

3) You have the right working and series, but it seems you subbed in x=2 instead of x= 0.2.

 

[Noober]

2. I did. I made an Excel thing to find the sums, and it converged to -0.617209760262333. The actual value of cos(63º) is 0.4539905. I can't figure out what's wrong...
Edit: I was doing it in degrees. I got it now.

3. Ah, thanks.

Can anyone help me with Part II B and Part IV?

 

[Gib Z]

IV) Let's do a and b first.

A) Just use the definition of a Maclaurin series, not hard to sub in.

B) Using that series for f(x), change the variable to t, integrate term by term.

 

[Noober]

I did that (see above pics), and it looks right. But part C doesn't work. If I set -2ak=1, and 2a²k/3=1, a comes out to be -1/3 and k=3/2. That doesn't work for the third term...

 

[Gib Z]

Try this approach:

$f(x)=6e^{-x/3}, f'(x)=-2e^{-x/3}$

$$kf'(ax) = -2k \cdot e^{-ax/3}$$

Now for the LHS, you should recognise that's the taylor series for e^x.

Now you just have to solve [itex]e^x = -2k \cdot e^{-ax/3}[/tex][/itex]

 

[Noober]

Ok, with that, I got k=-1/2, and a=-3. Right...?

 

[Gib Z]

Aww come on mate you don't need to be spoon fed like this :P Substituting those into the RHS of what I left you to solve, you can see that it is correct = ] but next time i want something more like:

" I got k=-1/2 and a=-3 which works out, yay!"

 

[Noober]

Ok, thanks for the help everyone. Well, actually, just Gib Z, but whatever :).