If -x-1 and y-1-x-x2-x3- - then write the value of d y-d x

We have y=1+x+x^2+x^3+...., =(1 x)^ 1 dy/dx=d/dx(1 x)^ 1 =( 1)(1 x)^ 2× ( 1) =(1 x)^ 2 =1/(1 x)^2 ...

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Expansions to Remove Indeterminate Form

If |x|<1 and ...

Question

If $| x | < 1$ and $y = 1 + x + x^{2} + x^{3} + . . . .,$ then write the value of $\frac{d y}{d x}$ .

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Solution

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\frac{d y}{d x}

y = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + . . . . \infty

\frac{d y}{d x}

y = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots,

\frac{d y}{d x} =

y = e^{log (1 + x + x^{2} + x^{3} + \dots)}

| x | < 1

\frac{d y}{d x}

y = 1 + \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + . . . . . \infty

| x | > 1

\frac{d y}{d x} =

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