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Math formulas: Special power series

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Powers of Natural Numbers

k=1nk=12n(n+1) \sum\limits_{k=1}^n k = \frac{1}{2}n(n+1)
k=1nk2=16n(n+1)(2n+1) \sum\limits_{k=1}^n k^2 = \frac{1}{6}n(n+1)(2n+1)
k=1nk3=14n2(n+1)2 \sum\limits_{k=1}^n k^3 = \frac{1}{4}n^2(n+1)^2

Special Power Series

11x=1+x+x2+x3+(for 1<x<1) \frac{1}{1-x} = 1 + x + x^2 +x^3 + \cdots \quad(\text{for } -1 < x < 1)
11+x=1x+x2x3+(for 1<x<1) \frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots \quad(\text{for } -1 < x < 1)
ex=1+x+x22!+x33!+ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
ln(1+x)=xx22+x33x44+(for 1<x<1) \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad (\text{for } -1 < x < 1)
sinx=xx33!+x55!x77!+ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
cosx=1x22!+x44!x66!+ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
tanx=xx33+2x51517x7315+(for π2<x<π2) \tan\,x = x - \frac{x^3}{3} + \frac{2x^5}{15} - \frac{17x^7}{315} + \cdots \quad \left(\text{for } -\frac{\pi}{2} < x < \frac{\pi}{2} \right)
sinhx=x+x33!+x55!+x77!+ \sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots
coshx=1+x22!+x44!+x66!+ \cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots

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