What is taylor cos(2x),pi ?

The answer to taylor cos(2x),pi is 1-2(x-pi)^2+2/3 (x-pi)^4-4/45 (x-pi)^6+2/315 (x-pi)^8+\ldots

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taylor cos(2x),pi

taylor

cos (2 x), π

1 - 2 (x - π)^{2} + \frac{2}{3} (x - π)^{4} - \frac{4}{45} (x - π)^{6} + \frac{2}{315} (x - π)^{8} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{0}{1 !} (x - π) + \frac{- 4}{2 !} (x - π)^{2} + \frac{0}{3 !} (x - π)^{3} + \frac{16}{4 !} (x - π)^{4} + \frac{0}{5 !} (x - π)^{5} + \frac{- 64}{6 !} (x - π)^{6} + \frac{0}{7 !} (x - π)^{7} + \frac{256}{8 !} (x - π)^{8} + \dots

Refine

= 1 - 2 (x - π)^{2} + \frac{2}{3} (x - π)^{4} - \frac{4}{45} (x - π)^{6} + \frac{2}{315} (x - π)^{8} + \dots

What is taylor cos(2x),pi ?
The answer to taylor cos(2x),pi is 1-2(x-pi)^2+2/3 (x-pi)^4-4/45 (x-pi)^6+2/315 (x-pi)^8+\ldots