Trig Trick In Solutions Manual

[Metronome]

In this solutions manual on PDF page 836, the answer to Section 12.1 #7 calculates the Fourier coefficient a_n to be (1 - (-1)^n)/(n^2 π^2), thus the cosine term in the sum should be (1 - (-1)^n)/(n^2 π^2) cos(nπx). However, in the final solution it is written as 2/(π^2) * (cos(2n + 1)πx)/((2n + 1)^2). How is this transition made?

 

[Steven G]

How much is (1 - (-1)^n) when n is even and when n is odd??

 

[Metronome]

0 and 2, respectively.

 

[Steven G]

0 and 2, respectively.

So update what you have with that information and show us what you get.

 

[Metronome]

0 for n even
2/(n^2 π^2) cos(nπx) for n odd

 

[Steven G]

Now go back to your last post and make 'n' odd.

 

[Metronome]

Now go back to your last post and make 'n' odd.

I'm not sure what you mean by that.

 

[Dr.Peterson]

The [MATH]n[/MATH] in the final version is not the same [MATH]n[/MATH] they used previously. They have replaced [MATH]n[/MATH] with [MATH]2n+1[/MATH]. It might be clearer if you first replace [MATH]n[/MATH] with [MATH]2k+1[/MATH] (which you can do when [MATH]n[/MATH] is odd), and then change [MATH]k[/MATH] to [MATH]n[/MATH], just renaming it.