The Fourier series to represent x-x^{2} for -\pi \leq x \leq \pi is given by x-x^{2}=\frac{a_{0}}{2}+\sum_{n=1}^{x} a_{n} \cos n x+\sum_{n-1}^{x} b_{n} \sin n x The value of a_{0} (round off to two decimal places), is________________.

GATE Civil Engineering (CE) 2020 Shift 2 Solved Paper

Question : 47 of 65

Marks: +1, -0

The Fourier series to represent

x-x^{2}

for

-π ≤ x ≤ π

is given by

x-x^{2}=\;{a_{0}}/{2}+∑↙{n=1}↖{x} a_{n} \cos n x+∑↙{n-1}↖{x} b_{n} \sin n x

The value of

a_{0}

(round off to two decimal places), is________________.

Solution:

Comparing the above equation with standard trigonometric farriers series representation,

f(x)=\;{a_{0}}/{2}+∑↙{n=1}↖{∞} a_{n} \cos n x+∑↙{n=1}↖{∞} b_{n} \sin n x

\;{a_{0}}/{2}=\;{1}/{T} ∫↙{0}↖{T} f(x) d x,-π < f(x) < π

\;{a_{0}}/{2}=\;{1}/{2 π} ∫↙{-π}↖{π}(x-x^{2}) d x

\;{a_{0}}/{2}=\;{1}/{2 π}[∫↙{-π}↖{π} x d x-∫↙{-π}↖{π} x^{2} d x]

\;{a_{0}}/{2}=\;{1}/{2 π}[[\;{x^{2}}/{2}]_{-π}^{π}-[\;{x^{3}}/{3}]_{-π}^{π}]

\;{a_{0}}/{2}=\;{1}/{2 π}[(\;{π^{2}}/{2}-\;{π^{2}}/{2})-(\;{π^{3}}/{3}-\;{-π^{3}}/{3})]

\;{a_{0}}/{2}=\;{1}/{2 π}[\;{-2 π 3}/{3}]

\;{a_{0}}/{2}=\;{-1}/{2 π} × \;{2 π^{3}}/{3}

\;{a_{0}}/{2}=\;{-π^{2}}/{3}

a_{0}=\;{-2}/{3} π^{2}=-6.58

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