- #1

- 230

- 0

## Homework Statement

show that A

_{k}will satisfy:

[tex]\left| A_{k} \right| \leq Mk^{-4} [/tex]

## Homework Equations

[tex] A_{k} = \frac{2}{L}\int^{L}_{0} \phi(x) sin \left( \frac{k \pi x}{L} \right) dx [/tex]

given

[tex] \phi(x) \in C^{4} ([0,L]) \and\ \phi^{(p)}(0) = \phi^{(p)}(L) = 0, p = 0, 1, 2, 3 [/tex]

in this notation, \phi is continuous and differentiable up to 4th derivative over [0,L], and (p) denotes the order of the derivative.

## The Attempt at a Solution

I differentiated 4 times using substitution, and came up with

[tex] \frac{L^{4}}{k^{4}\pi^{4}} \int^{L}_{0} \phi^{(4)}(x)sin \left( \frac{k \pi x}{L}\right) dx [/tex].

I was told using the Weierstrass M-test (or Comparison test?) would help find solution, but I'm not sure how to proceed.

Any comments or suggestions?