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321.
n-th terem of an arithmetic sequence
an=a1+(n1)da_n = a_1 + (n-1)\cdot d
322.
Sum of the first n terms
Sn=n2[2a1+(n1)d]S_n=\dfrac{n}{2}\left[2a_1 + (n-1)\cdot d\right]
323.
Sum of first n terms
Sn=n2(a1+an)S_n= \dfrac{n}{2}\left(a_1+a_n\right)
324.
n-th term
an=an1+an+12a_n=\dfrac{a_{n-1}+a_{n+1}}{2}
325.
n-th term
an=a1qn1a_n=a_1\cdot q^{n-1}
326.
Sum of first n terms
Sn=a1qn1q1S_n = a_1\frac{q^n-1}{q-1}
327.
Sum of first n terms
Sn=anqa1q1S_n = \dfrac{a_nq-a_1}{q-1}
328.
Common raito
q=ana1q=\dfrac{a_n}{a_1}
329.
11x=1+x+x2+x3+1<x<1\dfrac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \quad -1 < x < 1
330.
1(1+x)2=12x+3x24x3+1<x<1\dfrac{1}{(1+x)^2} = 1 - 2x + 3x^2 - 4x^3 + \cdots \quad -1 < x < 1
331.
1(1x)2=1+2x+3x2+4x3+1<x<1\dfrac{1}{(1-x)^2} = 1 + 2x + 3x^2 + 4x^3 + \cdots \quad -1 < x < 1
332.
1(1+x)3=13x+6x210x3+1<x<1\dfrac{1}{(1+x)^3} = 1 - 3x + 6x^2 - 10x^3 + \cdots \quad -1 < x < 1
333.
ex=1+x+x22!+x33!+e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots
334.
ax=1+xlna+(xlna)22!+(xlna)33!+a^x = 1 + x\,\ln a + \dfrac{(x\,\ln a)^2}{2!} + \dfrac{(x\,\ln a)^3}{3!} + \cdots
335.
ln(1+x)=xx22+x33x44+1<x1\ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots \quad -1 < x \leq 1
336.
sinx=xx33!+x55!x77!+\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots
337.
cosx=1x22!+x44!x66!+\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots
338.
arcsinx=x+12x33+1324x55+135246x77+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
339.
cothx=1x+x3x345+(1)n122nBnx2n1(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
340.
tanhx=xx33+2x515+(1)n122n(22n1)Bnx2n1(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}