calculus-I-prove-that-f-x-1-1-4x-n-0-2n-n-x-n-

Question Number 136723 by mnjuly1970 last updated on 25/Mar/21

\dots . c a l c u l u s (I) \dots . .

p r o v e t h a t ::

f (x) = \frac{1}{\sqrt{1 - 4 x}} \overset{? ? ?}{=} \underset{n = 0}{\sum^{\infty}} (\begin{matrix} 2 n \\ n \end{matrix}) x^{n}

Answered by Dwaipayan Shikari last updated on 25/Mar/21

\frac{1}{\sqrt{1 - 4 x}} = 1 + \frac{4 x}{2.1!} + \frac{{(4 x)}^{2}}{2^{2} . 2!} + \frac{{(4 x)}^{3}}{2^{3} . 3!} + \frac{{(4 x)}^{4}}{2^{4} . 4!} + \dots

= 1 + (\begin{matrix} 2 \\ 1 \end{matrix}) x^{1} + (\begin{matrix} 4 \\ 2 \end{matrix}) x^{2} + (\begin{matrix} 6 \\ 3 \end{matrix}) x^{3} + \dots = \underset{n = 0}{\sum^{\infty}} (\begin{matrix} 2 n \\ n \end{matrix}) x^{n}

C o n v e r g e s w h e n x < \frac{1}{4}

\sqrt{5} = 1 + \frac{4}{2.5.1!} + \frac{4^{2}}{5^{2} . 2^{2} . 2!} + . . a d i n f

Commented by mnjuly1970 last updated on 25/Mar/21

t h a n k s a l o t \dots