A) \[y+{{y}^{2}}+{{y}^{3}}+...to\,\infty \]
B) \[1+y+{{y}^{2}}-{{y}^{3}}+....to\,\infty \]
C) \[1-2y+3{{y}^{2}}-....to\,\infty \]
D) \[1+2y+3{{y}^{2}}+....to\,\infty \]
E) \[y+{{y}^{2}}+{{y}^{3}}-....to\,\infty \]
Correct Answer: C
Solution :
\[y=x+{{x}^{2}}+{{x}^{3}}+...\] \[\Rightarrow \] \[y=\frac{x}{1-x}\] \[\Rightarrow \] \[y-yx=x\] \[\Rightarrow \] \[x=\frac{y}{1+y}=y-{{y}^{2}}+{{y}^{3}}-....\] On differentiating w.r.t. y, we get \[\frac{dx}{dy}=1-2y+3{{y}^{2}}-.....\]
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