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

The answer to taylor (e^x-e^{-x})/2 is x+1/6 x^3+1/120 x^5+1/5040 x^7+1/362880 x^9+\ldots

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

taylor

\frac{e ^{x} - e ^{- x}}{2}

x + \frac{1}{6} x^{3} + \frac{1}{120} x^{5} + \frac{1}{5040} x^{7} + \frac{1}{362880} x^{9} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{1}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{1}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{1}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{1}{9 !} x^{9} + \dots

Refine

= x + \frac{1}{6} x^{3} + \frac{1}{120} x^{5} + \frac{1}{5040} x^{7} + \frac{1}{362880} x^{9} + \dots

\frac{d}{d x} (ln (e^{3 x^{2}} \cdot 4^{5 x}))

\frac{\partial}{\partial x} ((1 + x y)^{\frac{3}{2}})

x \to \infty lim (e^{- 4 x} cos (4 x))

\int_{0}^{10} x - 5 d x

\frac{\partial}{\partial x} (- x + 345 x + 10000)

What is taylor (e^x-e^{-x})/2 ?
The answer to taylor (e^x-e^{-x})/2 is x+1/6 x^3+1/120 x^5+1/5040 x^7+1/362880 x^9+\ldots