PYQs Based Test - 07 : Taylor and Fourier Series

The infinite series $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$ corresponds to

In the neighbourhood of z = 1, the function f(z) has a power series expansion of the form

𝑓(𝑧) = 1 + (1 − 𝑧) + (1 − 𝑧)² + ...∞

Then f(z) is

Let $f\left(x\right)=e^{x+x^2}$ for real x. From among the following, choose the Taylor series approximation of f(x) around x = 0, which includes all powers of x less than or equal to 3.

The Fourier series to represent x-x² for –π ≤ x ≤ π is given by $x-x^2=\frac{a_0}{2}+\sum _{n=1}^{\infty }{a}_{n}cosnx+\sum _{n=1}^{\infty }{b}_{n}sinnx$

The value of a₀ (round off to two decimal places), is

The Taylor series expansion of 3 sin x + 2 cos x is