The triangular wave is shown opposite. Because of the discontinuities in
the derivative, then the function has to be expressed as 3 pieces.
These are: f(t) =  4t/T  2 for T/2 £ t £ T/4 f(t) = 4t/T for T/4 £ t £ T/4 f(t) =  4t/T + 2 for T/4 £ t £ T/2 

Since the Function id odd a_{n} = 0 for all n and only the coefficients b_{n } occur in the Fourier Series for f(t). Expression to evaluate b_{n} is shown opposite. 

Substituting the Expressions for f(t) into that for b_{n} gives the result Þ 

The Expression for b_{n} can be simplified by replacing t with  t in the first integral. This gives Þ 

The first 2 integrals contain the term tsinnwt and may be integrated by parts using the result Þ 

Evaluating the integrals and using wT = 2p result in the expression opposite for b_{n}. 

The triangle wave can now be represented by the Fourier Series Þ. 
