taylor x^x

Solution

taylor

x^{x}

1 + (x - 1) + (x - 1)^{2} + \frac{1}{2} (x - 1)^{3} + \frac{1}{3} (x - 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( x ^{x} ) ( 1 )}{1 !} (x - 1) + \frac{\frac{d ^{2}}{d x ^{2}} ( x ^{x} ) ( 1 )}{2 !} (x - 1)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( x ^{x} ) ( 1 )}{3 !} (x - 1)^{3} + \dots

Evaluate Derivatives

= 1 + \frac{1}{1 !} (x - 1) + \frac{2}{2 !} (x - 1)^{2} + \frac{3}{3 !} (x - 1)^{3} + \frac{8}{4 !} (x - 1)^{4} + \dots

Refine

= 1 + (x - 1) + (x - 1)^{2} + \frac{1}{2} (x - 1)^{3} + \frac{1}{3} (x - 1)^{4} + \dots

(3 - x^{2})^{2}, x^{4}

(2 x + 3)^{1.4}

x \frac{d y}{d x} = - y + 2 x^{2} y

(2 x^{2} + 1) (x - 3)^{2}

\int 8 sin^{3} (x) d x

What is taylor x^x ?
The answer to taylor x^x is 1+(x-1)+(x-1)^2+1/2 (x-1)^3+1/3 (x-1)^4+\ldots