What is taylor e^{x^2}-1,0 ?

The answer to taylor e^{x^2}-1,0 is x^2+1/2 x^4+1/6 x^6+1/24 x^8+1/120 x^{10}+\ldots

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taylor e^{x^2}-1,0

taylor

e^{x^{2}} - 1, 0

x^{2} + \frac{1}{2} x^{4} + \frac{1}{6} x^{6} + \frac{1}{24} x^{8} + \frac{1}{120} x^{10} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \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} + \frac{0}{9 !} x^{9} + \frac{30240}{10 !} x^{10} + \dots

Refine

= x^{2} + \frac{1}{2} x^{4} + \frac{1}{6} x^{6} + \frac{1}{24} x^{8} + \frac{1}{120} x^{10} + \dots

x \to - 2 lim (8 - x^{3})

\int \frac{1}{cosh ^{2} ( x )} d x

x \to \frac{π}{2} + lim (\frac{1}{cos ( x )})

f (t) = t e^{- 3 t} cos (5 t)

\frac{d}{d x} (v) = \frac{1}{2 v}

What is taylor e^{x^2}-1,0 ?
The answer to taylor e^{x^2}-1,0 is x^2+1/2 x^4+1/6 x^6+1/24 x^8+1/120 x^{10}+\ldots