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(Solved): (a) (i) Prove that \( \frac{1}{1+x}=1-x+x^{2}-x^{3}+x^{4}-\cdots=\sum_{n=0}^{\infty}(-1)^{n} x^{n} ...

(a) (i) Prove that \( \frac{1}{1+x}=1-x+x^{2}-x^{3}+x^{4}-\cdots=\sum_{n=0}^{\infty}(-1)^{n} x^{n} \) for \( |x|<1 \)
(ii) Fi

(a) (i) Prove that \( \frac{1}{1+x}=1-x+x^{2}-x^{3}+x^{4}-\cdots=\sum_{n=0}^{\infty}(-1)^{n} x^{n} \) for \( |x|<1 \) (ii) Find a series expansion for \( \ln (1+x) \) by term by term integration of the series expansion obtained in (a)(i). (iii) Represent the integral \( \int_{0}^{x} \frac{\ln (1+t)}{t} d t \) as a power series expansion. Continued... 2 Find the Fourier series for the function \[ f(x)=\left\{\begin{array}{cc} \frac{\pi}{2}+x & -\pi \leq x \leq 0 \\ \frac{\pi}{2}-x & 0

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A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. F
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