Calculus Course/Series

SeriesEdit

A series is the sum of the terms of a sequence.

Finite sequences and series have defined first and last terms, whereas
infinite sequences and series continue indefinitely

The sum of terms of a sequence is a series. More precisely, if (x1, x2, x3, ...) is a sequence, one may consider the sequence of partial sums (S1, S2, S3, ...), with

S_n=x_1+x_2+\dots + x_n=\sum\limits_{i=1}^{n}x_i.

Sums of powersEdit

  • \sum_{m=1}^n m = 1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}\,\!
  • \sum_{m=1}^n m^2 = 1^2 + 2^2 + 3^2 + ... + n^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}  \,\!
  • \sum_{m=1}^n m^3 = 1^3 + 2^3 + 3^3 + ... + n^3 = \left[\frac{n(n+1)}{2}\right]^2 = \frac{n^4}{4} + \frac{n^3}{2} + \frac{n^2}{4} = \left(\sum_{m=1}^n m\right)^2\,\!
  • \sum_{m=1}^{n} m^{4} = = 1^4 + 2^4 + 3^4 + ... + n^4 = \frac{n(n+1)(2n+1)(3n^{2}+3n-1)}{30}=\frac{6 n^5+15 n^4+10 n^3-n}{30}\,\!
  • \sum_{m=0}^n m^s = \frac{(n+1)^{s+1}}{s+1} + \sum_{k=1}^s\frac{B_k}{s-k+1}{s\choose k}(n+1)^{s-k+1}\,\!
where B_k\, is the k\,th Bernoulli number, and B_1\, is negative.
  • \sum^{\infty}_{m=1} \frac{1}{m^2} = \frac{\pi^2}{6}\,\!
  • \sum^{\infty}_{m=1} \frac{1}{m^4} = \frac{\pi^4}{90}\,\!
  • \sum^{\infty}_{m=1} \frac{1}{m^{2n}} = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}
  • \sum_{m=1}^\infty m^{-s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = \zeta(s)\,\!
where \zeta(s)\, is the Riemann zeta function.

Power seriesEdit

Infinite sum (for |x| < 1) Finite sum
\sum_{m=0}^\infty x^m= \frac{1}{1-x}\,\! \sum_{m=0}^n x^m = \frac{1-x^{n+1}}{1-x} = 1+\frac{1}{r}\left(1-\frac{1}{(1+r)^n}\right) where r>0 and x=\frac{1}{1+r}.\,\!
\sum_{m=0}^\infty x^{2m}= \frac{1}{1-x^2}\,\!
\sum_{m=1}^\infty m x^m = \frac{x}{(1-x)^2}\,\! \sum_{m=1}^n m x^m = x\frac{1-x^n}{(1-x)^2} - \frac{n x^{n+1}}{1-x}\,\!
\sum_{m=1}^{\infty} m^2 x^m =\frac{x(1+x)}{(1-x)^3}\,\! \sum_{m=1}^n m^2 x^m = \frac{x(1+x-(n+1)^2x^n+(2n^2+2n-1)x^{n+1}-n^2x^{n+2})}{(1-x)^3} \,\!
\sum_{m=1}^{\infty} m^3 x^m =\frac{x(1+4x+x^2)}{(1-x)^4}\,\!
\sum_{m=1}^{\infty} m^4 x^m =\frac{x(1+x)(1+10x+x^2)}{(1-x)^5}\,\!
\sum_{m=1}^{\infty} m^k x^m
 = \operatorname{Li}_{-k}(x),\,\! where Lis(x) is the polylogarithm of x.

Simple denominatorsEdit

  • \sum^{\infty}_{m=1} \frac{x^m}m = \ln\left(\frac{1}{1-x}\right) \quad\mbox{ for } |x| < 1 \!
  • \sum^{\infty}_{m=0} \frac{(-1)^m}{2m+1} x^{2m+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots = \arctan(x)\,\!
  • \sum^{\infty}_{m=0} \frac{x^{2m+1}}{2m+1} = \mathrm{arctanh} (x) \quad\mbox{ for } |x| < 1\,\!

Factorial denominatorsEdit

Many power series which arise from Taylor's theorem have a coefficient containing a factorial.

  • \sum^{\infty}_{m=0} \frac{x^m}{m!} = e^x
  • \sum^{\infty}_{m=0} \frac{(-1)^m}{m!} x^{m} = \frac{1}{e^{x}}
  • \sum^{\infty}_{m=0} \frac{(-1)^m}{(2m+1)!} x^{2m+1}=  x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sin x
  • \sum^{\infty}_{m=0} \frac{(-1)^m}{(2m)!} x^{2m} =  1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \cos x
  • \sum^{\infty}_{m=0} \frac{x^{2m+1}}{(2m+1)!} = \sinh x
  • \sum^{\infty}_{m=0} \frac{x^{2m}}{(2m)!} = \cosh x

Modified-factorial denominatorsEdit

  • \sum^{\infty}_{m=0} \frac{(2m)!}{4^m (m!)^2 (2m+1)} x^{2m+1} = \arcsin x\quad\mbox{ for } |x| < 1\!
  • \sum^{\infty}_{m=0} \frac{(-1)^m (2m)!}{4^m (m!)^2 (2m+1)} x^{2m+1} = \mathrm{arcsinh}(x) \quad\mbox{ for } |x| < 1\!

Binomial seriesEdit

Geometric series:

  • 
 (1+x)^{-1} = 
 \begin{cases} 
   \displaystyle  \sum_{m=0}^\infty (-x)^m     &  |x|<1  \\
   \displaystyle  \sum_{m=1}^\infty -(x)^{-m}  &  |x|>1  \\
 \end{cases}

Binomial Theorem:

  • 
 (a+x)^n = 
 \begin{cases} 
   \displaystyle \sum_{m=0}^\infty \binom{n}{m} a^{n-m} x^m  &  |x| \! < \! |a|  \\
   \displaystyle \sum_{m=0}^\infty \binom{n}{m} a^m x^{n-m}  &  |x| \! > \! |a|  \\
 \end{cases}


  • (1+x)^\alpha = \sum_{m=0}^\infty {\alpha \choose m} x^m\quad\mbox{ for all } |x| < 1 \mbox{ and all complex } \alpha\!
with generalized binomial coefficients
{\alpha\choose n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}\!

Square root:

  • \sqrt{1+x} = \sum_{m=0}^\infty \frac{(-1)^m(2m)!}{(1-2m)m!^24^m}x^m \quad\mbox{ for } |x|<1\!

Miscellaneous:

Bernoulli NumbersEdit

Harmonic NumbersEdit

Binomial coefficientsEdit

  • \sum_{m=0}^n {n \choose m} = 2^n
  • \sum_{m=0}^n {n \choose m}a^{(n-m)} b^m = (a + b)^n
  • \sum_{m=0}^n (-1)^i{n \choose m} = 0
  • \sum_{m=0}^n {m \choose k} = { n+1 \choose k+1 }
  • \sum_{m=0}^n {k+m \choose m} = { k + n + 1 \choose n }
  • \sum_{m=0}^r {r \choose m}{s \choose n-m} = {r + s \choose n}

Trigonometric functionsEdit

Sums of sines and cosines arise in Fourier series.

  • \sum_{m=1}^n \sin\left(\frac{m\pi}{n}\right) = 0
  • \sum_{m=1}^n \cos\left(\frac{m\pi}{n}\right) = 0

UnclassifiedEdit

  • \sum_{m=b+1}^{\infty} \frac{b}{m^2 - b^2} = \frac{1}{2} H_{2b}
  • \sum^{\infty}_{m=1} \frac{y}{m^2+y^2} = -\frac{1}{2y}+\frac{\pi}{2}\coth(\pi y)


Last modified on 26 February 2011, at 19:48