# Thread: Weirstrass M-test (convergence) and derivative

1. ## Weirstrass M-test (convergence) and derivative

Hi,

I would like to know how to show the following:

Let f(x)= summation of an (note that n is subscript) * (Sin nx)
so f(x) = summation (an)(Sin nx).
If the series summation of n(an) is absolutely convergent, show that f is differentiable and f'(x)= summation of n(an)(cos nx).

I would also like to know what theorem on series of derivatives should we apply.

2. Do you go to Concordia university by any chance?

3. Originally Posted by zxcv
Hi,

I would like to know how to show the following:

Let f(x)= summation of an (note that n is subscript) * (Sin nx)
so f(x) = summation (an)(Sin nx).
If the series summation of n(an) is absolutely convergent, show that f is differentiable and f'(x)= summation of n(an)(cos nx).

I would also like to know what theorem on series of derivatives should we apply.
What is the conditions for a series to be differentiable?

4. Originally Posted by zxcv
Hi,

I would like to know how to show the following:

Let f(x)= summation of an (note that n is subscript) * (Sin nx)
so f(x) = summation (an)(Sin nx).
If the series summation of n(an) is absolutely convergent, show that f is differentiable and f'(x)= summation of n(an)(cos nx).

Let $\displaystyle F_k(x) = \sum_{i=0}^k a_n sin(nx)$
Clearly $\displaystyle F'_k(x) = \sum_{i=0}^k na_ncos(nx)$
And there's a theorem that says if $\displaystyle {F_n}$ converges uniformly to F and $\displaystyle F'_n$ converges uniformly to G, then F' = G....