What is taylor 1-cos(x) ?

The answer to taylor 1-cos(x) is 1/2 x^2-1/24 x^4+1/720 x^6-1/40320 x^8+1/3628800 x^{10}+\ldots

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taylor 1-cos(x)

taylor

1 - cos (x)

\frac{1}{2} x^{2} - \frac{1}{24} x^{4} + \frac{1}{720} x^{6} - \frac{1}{40320} x^{8} + \frac{1}{3628800} x^{10} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{0}{1 !} x + \frac{1}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{- 1}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{1}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{- 1}{8 !} x^{8} + \frac{0}{9 !} x^{9} + \frac{1}{10 !} x^{10} + \dots

Refine

= \frac{1}{2} x^{2} - \frac{1}{24} x^{4} + \frac{1}{720} x^{6} - \frac{1}{40320} x^{8} + \frac{1}{3628800} x^{10} + \dots

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\int_{0}^{1} \frac{- 4 y}{e ^{2 y}} d y

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What is taylor 1-cos(x) ?
The answer to taylor 1-cos(x) is 1/2 x^2-1/24 x^4+1/720 x^6-1/40320 x^8+1/3628800 x^{10}+\ldots