Question

what is 2x+3x^2+4x^3​

Answer 1

Step-by-step explanation:

Here is a quite different approach. [math]\text{Let }S=1+2x+3x^2+4x^3+\dots\infty[/math] [math]\text{Integrating, we get,}[/math] [math]\int Sdx=1+x+x^2+x^3+x^4+\dots ...

Answer 2

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$\bf\implies 1+2x+3x²+4x³....$

$\bf\implies Let S ∞ =1+2x+3x²+4x +...∞ ...(1)$

$\bf\implies Multiplying equation (1) by x, we get$

$\bf\implies xS∞=x+2x²+3x³+4x⁴ +...∞ ...(2)$

$\bf\implies Subtracting (2) from (1); we get$

$\bf\implies S∞−xS∞=(1+2x+3x²+4x³+...∞)−(x+2x²+3x³ +4x⁴ +...∞)$

$\bf\implies (1−x)S∞=1+x+x² +x³ +...∞$

$\implies Notice that the series 1+x+x² +x³+...∞ is geometric series with the first term a=1 and the common ratio r=x.$

Now, use the formula for the sum of an infinite geometric series.

⇒(1−x)S∞= $\frac{1}{1-x}$ ,for ∣x∣<1

⇒S∞=$\frac{1}{1-x²}$