Series till Ad Infinitum part I I II

Calculus Level 4

1 + 1 2 − 1 4 ⋅ 2 ! − 1 8 ⋅ 3 ! + 1 16 ⋅ 4 ! + 1 32 ⋅ 5 ! − ⋯ \large 1+ \dfrac{1}{2} - \dfrac{1}{4\cdot 2!} -\dfrac{1}{8\cdot 3!} + \dfrac{1}{16\cdot 4!} +\dfrac{1}{32\cdot 5!} -\cdots

sin ⁡ ( 1 4 ) + cos ⁡ ( 1 4 ) \sin\left(\frac{1}{4}\right ) + \cos \left(\frac{1}{4}\right ) sin ⁡ ( 1 2 ) + cos ⁡ ( 1 2 ) \sin\left(\frac{1}{2}\right ) + \cos\left(\frac{1}{2}\right ) tan ⁡ ( 1 2 ) + cot ⁡ ( 1 2 ) \tan\left(\frac{1}{2}\right ) + \cot \left(\frac{1}{2}\right ) sin ⁡ ( 1 3 ) + cos ⁡ ( 1 3 ) \sin \left(\frac{1}{3}\right ) +\cos\left(\frac{1}{3}\right )

Kunal Gupta by Kunal Gupta , 23, India

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1 solution

Caleb Townsend
Mar 14, 2015

The series is the sum of the power series of sin ⁡ ( x ) \sin(x) and cos ⁡ ( x ) \cos(x) at x = 1 2 . x = \frac{1}{2}. So the sum is sin ⁡ ( 1 2 ) + cos ⁡ ( 1 2 ) \boxed{\sin(\frac{1}{2}) + \cos(\frac{1}{2})}

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