What is taylor e^{(-x^2)} ?

The answer to taylor e^{(-x^2)} is 1-x^2+1/2 x^4-1/6 x^6+1/24 x^8+\ldots

taylor e^{(-x^2)}

taylor

e^{(- x^{2})}

1 - x^{2} + \frac{1}{2} x^{4} - \frac{1}{6} x^{6} + \frac{1}{24} x^{8} + \dots

Solution steps

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( e ^{- x^{2}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{- x^{2}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{- x^{2}} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 1 + \frac{0}{1 !} x + \frac{- 2}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{12}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 120}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{1680}{8 !} x^{8} + \dots

Refine

= 1 - x^{2} + \frac{1}{2} x^{4} - \frac{1}{6} x^{6} + \frac{1}{24} x^{8} + \dots

\int 5^{- x - 2} d x

\int \frac{1}{( x ^{2} + 4 x + 5 ) ^{2}} d x

\int \frac{1}{( x + 4 ) ( x + 8 )} d x

s l o p e in t erce pt (3, - 2), (7, 0)

x \to 1 lim (\frac{x - 2}{x ^{2} - 1})

What is taylor e^{(-x^2)} ?
The answer to taylor e^{(-x^2)} is 1-x^2+1/2 x^4-1/6 x^6+1/24 x^8+\ldots