1+frac{1}{4}+frac{1 cdot 3}{4 cdot 8}+frac{1 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The general binomial expansion for $(1-x)^{-n}$ is given by $(1-x)^{-n} = 1 + nx + \frac{n(n+1)}{2!}x^2 + \frac{n(n+1)(n+2)}{3!}x^3 + \ldots$ For

Vedclass · Accepted Answer

$\sqrt{2}$

Vedclass · Answer

(A) The general binomial expansion for $(1-x)^{-n}$ is given by $(1-x)^{-n} = 1 + nx + \frac{n(n+1)}{2!}x^2 + \frac{n(n+1)(n+2)}{3!}x^3 + \ldots$ For $n = \frac{1}{2}$,we have $(1-x)^{-1/2} = 1 + \frac{1}{2}x + \frac{\frac{1}{2}(\frac{3}{2})}{2!}x^2 + \frac{\frac{1}{2}(\frac{3}{2})(\frac{5}{2})}{3!}x^3 + \ldots$ $(1-x)^{-1/2} = 1 + \frac{1}{2}x + \frac{1 \cdot 3}{2^2 \cdot 2!}x^2 + \frac{1 \cdot 3 \cdot 5}{2^3 \cdot 3!}x^3 + \ldots$ $(1-x)^{-1/2} = 1 + \frac{x}{2} + \frac{1 \cdot 3}{8}x^2 + \frac{1 \cdot 3 \cdot 5}{48}x^3 + \ldots$ Comparing this with the given series $1 + \frac{1}{4} + \frac{1 \cdot 3}{32} + \ldots$,we set $\frac{x}{2} = \frac{1}{4}$,which gives $x = \frac{1}{2}$. Substituting $x = \frac{1}{2}$ into the expansion: $(1 - \frac{1}{2})^{-1/2} = (\frac{1}{2})^{-1/2} = \sqrt{2}$. Thus,the sum of the series is $\sqrt{2}$.

$1+\frac{1}{4}+\frac{1 \cdot 3}{4 \cdot 8}+\frac{1 \cdot 3 \cdot 5}{4 \cdot 8 \cdot 12}+\ldots$ is equal to

Similar Questions

If the bond length and dipole moment of a diatomic molecule are $1.25 \ \mathring{A}$ and $1.0 \ D$ respectively,what is the per cent ionic character of the bond?

The balancing length for a cell is $560 \, cm$ in a potentiometer experiment. When an external resistance of $10 \, \Omega$ is connected in parallel to the cell, the balancing length changes by $60 \, cm$. The internal resistance of the cell in ohms is:

Which of the following statements is $NOT$ a part of $IPM$ (Integrated Pest Management)?

The slopes of the focal chords of the parabola $y^2=32x$,which are tangents to the circle $x^2+y^2=4$,are

Two identical charged spherical drops, each of capacitance $C$, merge to form a single drop. The resultant capacitance is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module