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321.
n-th terem of an arithmetic sequence
an=a1+(n−1)⋅da_n = a_1 + (n-1)\cdot d
Arithmetic Series
322.
Sum of the first n terms
Sn=n2[2a1+(n−1)⋅d]S_n=\dfrac{n}{2}\left[2a_1 + (n-1)\cdot d\right]
Arithmetic Series
323.
Sum of first n terms
Sn=n2(a1+an)S_n= \dfrac{n}{2}\left(a_1+a_n\right)
Arithmetic Series
324.
n-th term
an=an−1+an+12a_n=\dfrac{a_{n-1}+a_{n+1}}{2}
Arithmetic Series
325.
n-th term
an=a1⋅qn−1a_n=a_1\cdot q^{n-1}
Geometric Series
326.
Sum of first n terms
Sn=a1qn−1q−1S_n = a_1\frac{q^n-1}{q-1}
Geometric Series
327.
Sum of first n terms
Sn=anq−a1q−1S_n = \dfrac{a_nq-a_1}{q-1}
Geometric Series
328.
Common raito
q=ana1q=\dfrac{a_n}{a_1}
Geometric Series
329.
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
330.
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
331.
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
332.
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
333.
ex=1+x+x22!+x33!+⋯e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots
Power series
334.
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
335.
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
336.
sin⁡x=x−x33!+x55!−x77!+⋯\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots
Power series
337.
cos⁡x=1−x22!+x44!−x66!+⋯\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots
Power series
338.
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
339.
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
340.
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

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