checking a formula

1 files changed, 4 insertions, 1 deletions

diff --git a/ClassJuly5.page b/ClassJuly5.page
index 8a1ac2c..9af2b85 100644
--- a/ClassJuly5.page
+++ b/ClassJuly5.page
@@ -51,7 +51,7 @@ $$ a(t) = e^{-\lambda^2 t} $$
 
 
 Next we solve for $b$.  It satisfies the equation
-$$ \frac{d^2b}{dx^2} = -\lambda b $$
+$$ \frac{d^2b}{dx^2} = -\lambda^2 b $$
 However we have to be a bit more careful in picking our solutions because $b$ is supposed to satisfy the boundary conditions
 $$ b(0) = b(L) = 0$$
 To satisfy $b(0) = 0$, we must take $b$ to be (a constant multiple of) a sine function:
@@ -73,3 +73,6 @@ In fact, every function of the kind described above does have a Fourier sine exp
 
 How do we know that the Fourier series of a square wave or sawtooth function converges?
 
+The answer to this question depends greatly on the type of convergence desired.  Aside from the convergence we already proved, the next easiest type of convergence is $L^2$ or root-mean-square convergence.  The formal statement is that
+
+$$ \lim_{N \to \infty} \sqrt{\int_0^{2\pi} | \sum_{n = - N}^N c_n e^{in\theta} - f(\theta) |^2} = 0 $$ 
\ No newline at end of file