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1.
11−x=1+x+x2+x3+⋯−1<x<1\dfrac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \quad -1 < x < 1
Power series
2.
1(1+x)2=1−2x+3x2−4x3+⋯−1<x<1\dfrac{1}{(1+x)^2} = 1 - 2x + 3x^2 - 4x^3 + \cdots \quad -1 < x < 1
Power series
3.
1(1−x)2=1+2x+3x2+4x3+⋯−1<x<1\dfrac{1}{(1-x)^2} = 1 + 2x + 3x^2 + 4x^3 + \cdots \quad -1 < x < 1
Power series
4.
1(1+x)3=1−3x+6x2−10x3+⋯−1<x<1\dfrac{1}{(1+x)^3} = 1 - 3x + 6x^2 - 10x^3 + \cdots \quad -1 < x < 1
Power series
5.
ex=1+x+x22!+x33!+⋯e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots
Power series
6.
ax=1+x ln⁡a+(x ln⁡a)22!+(x ln⁡a)33!+⋯a^x = 1 + x\,\ln a + \dfrac{(x\,\ln a)^2}{2!} + \dfrac{(x\,\ln a)^3}{3!} + \cdots
Power series
7.
ln⁡(1+x)=x−x22+x33−x44+⋯−1<x≤1\ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots \quad -1 < x \leq 1
Power series
8.
sin⁡x=x−x33!+x55!−x77!+⋯\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots
Power series
9.
cos⁡x=1−x22!+x44!−x66!+⋯\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots
Power series
10.
arcsin⁡x=x+12x33+1⋅32⋅4x55+1⋅3⋅52⋅4⋅6x77+⋯−1<x<1\arcsin x = x + \dfrac{1}{2}\dfrac{x^3}{3} + \dfrac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \dfrac{1\cdot3\cdot5}{2\cdot4\cdot6}\frac{x^7}{7} + \cdots \quad -1 < x <1
Power series
11.
coth⁡x=1x+x3−x345+⋯(−1)n−122nBnx2n−1(2n)!+⋯0<∣x∣<π\coth x = \dfrac{1}{x} + \dfrac{x}{3} - \dfrac{x^3}{45} + \cdots \dfrac{(-1)^{n-1} 2^{2n}B_nx^{2n-1}}{(2n)!} + \cdots \quad 0 < |x| < \pi
Power series
12.
tanh⁡x=x−x33+2x515+⋯(−1)n−122n(22n−1)Bnx2n−1(2n)!+⋯∣x∣<π2\tanh x = x - \dfrac{x^3}{3} + \dfrac{2x^5}{15} + \cdots \dfrac{(-1)^{n-1} 2^{2n}\left(2^{2n}-1\right)B_nx^{2n-1}}{(2n)!} + \cdots \quad |x| < \frac{\pi}{2}
Power series
13.
cosh⁡x=1+x22!+x44!+⋯\cosh x = 1 + \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + \cdots
Power series
14.
sinh⁡x=x+x33!+x55!+⋯\sinh x = x + \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + \cdots
Power series
15.
arccos⁡x=π2−arcsin⁡x=π2−(x+12x33+1⋅32⋅4x55+⋯ )−1<x<1\arccos x = \dfrac{\pi}{2} - \arcsin x = \dfrac{\pi}{2} - \left(x + \dfrac{1}{2}\dfrac{x^3}{3} + \dfrac{1\cdot3}{2\cdot4}\frac{x^5}{5}+\cdots \right) \quad -1 < x < 1
Power series
16.
ln⁡(1+x)=(x−1x)+12(x−1x)2+13(x−1x)3+⋯x≥12\ln(1+x) = \left(\dfrac{x-1}{x}\right) + \frac{1}{2}\left(\dfrac{x-1}{x}\right)^2 + \dfrac{1}{3}\left(\dfrac{x-1}{x}\right)^3 + \cdots \quad x \geq \frac{1}{2}
Power series
17.
cot⁡x=1x−x3−x345−⋯−22nBnx2n−1(2n)!0<x<π\cot x = \dfrac{1}{x} - \dfrac{x}{3} - \dfrac{x^3}{45} - \cdots - \frac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi
Power series
18.
1(1−x)3=1+3x+6x2+10x3+⋯−1<x<1\dfrac{1}{(1-x)^3} = 1 + 3x + 6x^2 + 10x^3 + \cdots \quad -1 < x < 1
Power series
19.
11+x=1−12x+1⋅32⋅4x2−1⋅3⋅52⋅4⋅6x3+⋯−1<x≤1\dfrac{1}{\sqrt{1+x}} = 1 - \dfrac{1}{2}x + \dfrac{1\cdot 3}{2\cdot 4}x^2 - \dfrac{1\cdot 3 \cdot 5}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1
Power series
20.
1+x=1+12x−12⋅4x2+1⋅32⋅4⋅6x3+⋯−1<x≤1\sqrt{1+x} = 1 + \dfrac{1}{2}x - \dfrac{1}{2\cdot 4}x^2 + \dfrac{1\cdot 3}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1
Power series

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