Representation of functions as power series question

• Jul 28th 2005, 02:57 PM
augury
Representation of functions as power series question
I am having a bit of difficulty with the representation of functions as power series. I thought I understand the basic concept and was able to do the first 15 or so problems in Single Variable Calculus by James Stewart (3rd ed). However, the following is one of my homework questions.

Let fn = ((sin nx) / n2). Show that the series Σ fn(x) converges for all values of x but the series of derivatives Σ f ‘n (x) diverges when x = 2n Π (PI), n is an integer. For what values of x does the series Σ f “n (x) converge?

I asked the only other student in the class I know and he couldn't do it either. So any help would be greatly appreciated.

Andrew
• Jul 28th 2005, 08:30 PM
hpe
Quote:

Originally Posted by augury
Let $f_n = \frac{\sin n x}{n^2}$. Show that the series $\sum_{n=1}^\infty f_n(x)$ converges for all values of x

Use the series comparison theorem to prove absolute convergence for all x. Compare with a p-series. I'll leave it to you to figure out which p to use :)
Quote:

but the series of derivatives $\sum_{n=1}^\infty f'_n (x)$ diverges when $x = 2k\pi$, k is an integer.
Note that I have changed n to k. It's possible that Stewart has a misprint here. In any case that is what is intended.

First find a formula for $f'_n(x)$. Then evaluate it at $x=2k\pi$ (it's the same value at $x = 0, \, x = 2\pi, \, x = 4\pi , \dots$). Then write down the resulting series. You should then have a p-series that is known to diverge.
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For what values of x does the series $\sum_{n=1}^\infty f"_n (x)$ converge?
Find a formula for $f"_n (x)$. Then use the series divergence test. The series diverges at "most" points, but there are infinitely many values where the series converges; $x=\pi$ is one of them. Proving the convergence at all these points should turn out to be completely trivial :)
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I asked the only other student in the class I know and he couldn't do it either. So any help would be greatly appreciated.
How about asking the prof? That's what he is getting paid for. :)

Btw, this is not a power series. It's what is called a trigonometric series.
• Jul 28th 2005, 10:26 PM
augury
Thank you for your reply. I will attempt the problem with what you have given me.