x-1-3-1-x-15-Find-the-limit-that-does-not-inclued-the-variable-x-in-the-opening-of-the-binomial-

Question Number 148333 by mathdanisur last updated on 27/Jul/21

{(\sqrt[3]{x} + \frac{1}{\sqrt{x}})}^{15}

F i n d t h e l i m i t t h a t d o e s n o t i n c l u e d

t h e v a r i a b l e x i n t h e o p e n i n g o f t h e

b i n o m i a l .

Answered by qaz last updated on 27/Jul/21

(\begin{matrix} 15 \\ 6 \end{matrix})

Commented by mathdanisur last updated on 27/Jul/21

H o w S i r

Answered by mindispower last updated on 27/Jul/21

{(a + b)}^{n} = \underset{k = 0}{\sum^{n}} C_{n}^{k} a^{k} b^{n - k}

a = x^{\frac{1}{3}}, x^{- \frac{1}{2}}, n = 15

= \underset{k = 0}{\sum^{n}} C_{n}^{k} x^{\frac{k}{3}} x^{- \frac{1}{2} (15 - k)}

\Rightarrow \frac{k}{3} - \frac{1}{2} (15 - k) = 0

\Rightarrow 2 k - 45 + 3 k = 0

k = 9

t h e c o n s t a n t e i s C_{9}^{15} = C_{15 - 9}^{15} = C_{6}^{15}

Commented by mathdanisur last updated on 27/Jul/21

T h a n k y o u S e r, a n s w e r : 9 . ?

Commented by qaz last updated on 27/Jul/21

\sqrt[3]{x} + \frac{1}{\sqrt{x}} = x^{\frac{1}{3}} + x^{- \frac{1}{2}}

{\begin{cases} \frac{1}{3} a - \frac{1}{2} b = 0 \\ a + b = 15 \end{cases}

\Rightarrow {\begin{cases} a = 9 \\ b = 6 \end{cases}

So canstant coefficient is (\begin{matrix} 15 \\ 6 \end{matrix}) .

Commented by mathdanisur last updated on 27/Jul/21

T h a n k y o u S i r