1. Taylors and MacLaurin's Series

Q.

How to find the Maclaurin series of $f\left(x\right)=\cos \left(x\right)$

Q. Taylor's series expansion of $f\left(z\right)=\frac{z-1}{z+1}$ about the point $z=0$ is

1. $1+2(z+z^2+z^3\dots \dots )$
2. $-1-2(z-z^2+z^3\dots \dots )$
3. $-1+2(z-z^2+z^3\dots \dots )$
4. None

Q.

If the coefficients of $(3r{)}^{\text{th}}$ and $(r+2{)}^{\text{th}}$ terms in the expansion of $(1+x{)}^{2n}$ are equal $(r>1,n>2)$, then

1. $n=r$

2. $n=r+1$

3. $n=2r$

4. $n=2r-1$

Q. The coefficient of $(x-1{)}^2$ in the Taylor series expansion of $f\left(x\right)=x{e}^x,\left(eϵR\right)$ about the point x = 1 is

1. $\frac{e}{2}$
   
2. 2e
3. $\frac{3e}{2}$
   
4. 3e

Q. For the function $e^{-x}$, the linear approximation around x = 3 is

1. $e^{-3}$
2. $(2-x)e^{-3}$
3. $(4-x)e^{-3}$
4. None of the above