How do I prove my Fejer kernel definition is equivalent? In my notes, the definition of the Fejer kernel is 
$$
F_{n} = \sum_{j=-N}^{N} \left(1 - \frac{|j|}{N+1}\right) e^{ijt}.
$$
But in most of the reference material I come across online, it is immediately defined as the average of the Dirichlet kernels
$$
F_{N} = \frac{1}{N+1} \left(D_{0} + \dots + D_{N}\right).
$$
I've tried equating these two definitions by expanding $F_{n}$'s $e^{ijt}$ and using some trigonometry to get something looking like the $\sin$ representation of the Dirichlet kernel but it has not been going well. 
Is there a simple way to prove that these two definitions are equivalent?

$\begin{array}{rlll}F_N(y)~&=~
\frac{1}{N+1}\sum_{n=0}^ND_n(y)=\frac{1}{N+1}\sum_{n=0}^N\sum_{k=-n}^ne^{iky}
\\ &=~ \frac{1}{N+1}\sum_{m=-N}^Ne^{imy}\#\{k \in [0,N]:m \in [-k,k]\}
\\ &=~ \frac{1}{N+1}\sum_{m=-N}^Ne^{imy}(N+1-|m|)=\sum_{m=-N}^Ne^{imy}(1-\frac{|m|}{N+1})
\end{array}$
