still testing

1 files changed, 5 insertions, 6 deletions

diff --git a/Fourier Series.page b/Fourier Series.page
index fdb70fc..a050d70 100644
--- a/Fourier Series.page
+++ b/Fourier Series.page
@@ -1,5 +1,3 @@
-<b>Lecture on Fourier Series:</b>  
-  
 ##Why Fourier series possible?</b>
 
 We first begin with a few basic identities on the size of sets. Show that the set of possible functions representing sets is not larger than the set of available functions?
@@ -8,10 +6,11 @@ We first begin with a few basic identities on the size of sets. Show that the se
 To show that Fourier series is plausible, let us consider some arbitrary trignometric functions and see if it is possible to express them as the sum of sines and cosines:  
  $1.  \cos(2x) = 1 - 2 \sin^2(x)$
 
-\begin{array}
-x = 1 \\
-c = 2
-\end{array}
+$$\begin{array}{ccl} 
+e^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\
+ & = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\
+ & = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\
+ & = & \cos y+i\sin y\end{array}$$
   
 ##What is the Fourier series actually?</b>