MCQ
$1 + n\left( {1 - \frac{1}{x}} \right) + \frac{{n(n + 1)}}{{2!}}{\rm{ }}{\left( {1 - \frac{1}{x}} \right)^2} + .....\infty $ = . . . .
- ✓${x^n}$
- B${x^{ - n}}$
- C${\left( {1 - \frac{1}{x}} \right)^n}$
- Dએકપણ નહીં.
${(1 + x)^n} = {\,^n}{C_0} + {\,^n}{C_1}x + {\,^n}{C_2}{x^2} + .....\infty $
If $x$ is replace by $ - \left( {1 - \frac{1}{x}} \right)$ and $n$ is $ - n$, then expression becomes ${\left[ {1 - \left( {1 - \frac{1}{x}} \right)} \right]^{ - n}}.$
$ = 1 + ( - n)\,\left[ { - \left( {1 - \frac{1}{x}} \right)} \right] + \frac{{( - n)( - n - 1)}}{{2!}}{\left[ { - \left( {1 - \frac{1}{x}} \right)} \right]^2} + ...$
or ${x^n} = 1 + n\left( {1 - \frac{1}{x}} \right) + \frac{{n(n + 1)}}{{2!}}{\left( {1 - \frac{1}{x}} \right)^2} + ....$
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વિધાન $1$ : $z$ એ વાસ્તવિક સંખ્યા છે.
વિધાન $2$ : $z$ નો મુખ્ય કોણાંક $\frac{\pi }{3}$ છે.