Answer to Question #224578 in Calculus for Modrick

Question

For the Fourier series in Problem 1, deduce a

series for

π

4

at the point where x =

π

2

Answer 1

For the Fourier series in Problem 1, deduce a series for $\dfrac{\pi}4$ at the point where $x= \dfrac{\pi}2$

$f(x)= \dfrac{a_0}2+ \Sigma a_n \sin \dfrac{nπx}{2}+ \Sigma b_n\cos\dfrac{nπx}{2}$

but b_n = 0

Hence, $f(x)= \dfrac{a_0}2+ \Sigma a_n \sin \dfrac{nπx}{2}$

When $\dfrac{\pi}2$, f(x) = 2, hence;

$2 = \dfrac8{\pi}\sin\dfrac{\pi}2+\dfrac8{3\pi}\sin\dfrac{3\pi}2+ \dfrac8{5\pi}\sin\dfrac{5\pi}2+...$

i.e

$2 =\dfrac8{\pi}(1-\dfrac13+\dfrac15-\dfrac17+...)$

$\dfracπ4 = 1-\dfrac13+\dfrac15-\dfrac17+...$