1-0- - 2-1- - 3-2- - 4-3- - - to - is equal to

The correct option is C 2e e^x=1+x+x^2/2!+x^3/3!+...∞ Multiplying by x on both the sides. xe^x=x+x^2+x^3/2!+x^4/3!+..∞ Differentiating with respec ...

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Standard X

Mathematics

Roots of Quadratic Equation

1/0! + 2/1! +...

Question

$\frac{1}{0!} + \frac{2}{1!} + \frac{3}{2!} + \frac{4}{3!} + . . .$ to $\infty$ is equal to

4 e

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3 e

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2 e

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none of these

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Solution

The correct option is C $2 e$
$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . . \infty$
Multiplying by x on both the sides.
$x e^{x} = x + x^{2} + \frac{x^{3}}{2!} + \frac{x^{4}}{3!} + . . \infty$
Differentiating with respect to x.
$e^{x} + x e^{x} = 1 + 2 x + \frac{3 x^{2}}{2!} + \frac{4 x^{3}}{3!} + . . . \infty$
Substituting, x=1, we get
$2 e = 1 + 2 + \frac{3}{2!} + \frac{4}{3!} + . . . \infty$
Hence the required answer is 2e.

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10!+21!+32!+43!+... to ∞ is equal to

The correct option is C 2eex=1+x+x22!+x33!+...∞Multiplying by x on both the sides.xex=x+x2+x32!+x43!+..∞Differentiating with respect to x.ex+xex=1+2x+3x22!+4x33!+...∞Substituting, x=1, we get2e=1+2+32!+43!+...∞Hence the required answer is 2e.

$\frac{1}{0!} + \frac{2}{1!} + \frac{3}{2!} + \frac{4}{3!} + . . .$ to $\infty$ is equal to