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in Determinants by (95 points)
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यदि \( y=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots \ldots .+\frac{x^{n}}{n !} \) है तो सिद्ध कीजिए कि \( \frac{d y}{d x}-y+\frac{x^{n}}{n !}=0 \).

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\(y = 1 + x + \frac{x^2}{2!} + ....+ \frac{x^n}{n!}\)

\(\frac{dy}{dx} = 1 + \frac{2x}{2!} + \frac{3x^2}{3!} + ....+ \frac{nx^{n-1}}{n(n-1)!}\)

⇒ \(\frac{dy}{dx} = 1 +x +\frac{x^2}{2!} + ...+ \frac{x^{n-1}}{(n-1)!}+ \frac{x^n}{n!} - \frac{x^n}{n!}\)

⇒ \(\frac{dy}{dx} = y - \frac{x^n}{n!}\)

⇒ \(\frac{dy}{dx} - y + \frac{x^n}{n!} = 0\)

Hence proved.

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