$f(x)=\left\{\begin{array}{cc}2 \sin \left(-\frac{\pi x}{2}\right), & \text { if } x<-1 \\ \left|a x^{2}+x+b\right|, & \text { if }-1 \leq x \leq 1 \\ \sin (\pi x), & \text { if } x>1\end{array}\right.$
If $f(x)$ is continuous on $R,$ then $a+b$ equals ..... .
$\Rightarrow f\left(1^{-}\right)=f(1)=f\left(1^{+}\right)$
$|a+1+b|=\lim _{x \rightarrow 1} \sin (\pi x)$
$|a+1+b|=0 \Rightarrow a+b=-1 ....(1)$
$\Rightarrow$ Also $f\left(-1^{-}\right)=f(-1)=f\left(-1^{+}\right)$
$\lim _{x \rightarrow-1} 2 \sin \left(\frac{-\pi x}{2}\right)=|a-1+b|$
$|a-1+b|=2$
Either $a-1+b=2$ or $a-1+b=-2$
$a + b =3 \ldots(2)$ or $a + b =-1 \ldots(3)$
from $(1)$ and $(2) \Rightarrow a+b=3=-1($ reject $)$
from $(1)$ and $(3) \Rightarrow a+b=-1$
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$(a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right)-\left(\frac{a^{4}+b^{4}}{4}\right)+\ldots$ is ..... .