List of mathematical series explained

See also: Summation.

This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.

00

is taken to have the value

1

\{x\}

denotes the fractional part of

x

Bn(x)

is a Bernoulli polynomial.

Bn

is a Bernoulli number, and here,
B
1=-1
2

.

En

is an Euler number.

\zeta(s)

is the Riemann zeta function.

\Gamma(z)

is the gamma function.

\psin(z)

is a polygamma function.

\operatorname{Li}s(z)

is a polylogarithm.

n\choosek

is binomial coefficient

\exp(x)

denotes exponential of

x

Sums of powers

See Faulhaber's formula.

m
\sum
k=0

kn-1=

Bn(m+1)-Bn
n
The first few values are:
m
\sumk=
k=1
m(m+1)
2
m
\sum
k=1
2=m(m+1)(2m+1)
6
k=
m3+
3
m2+
2
m
6
m
\sum
k=1
3 =\left[m(m+1)
2
k
2=m4
4
\right]+
m3+
2
m2
4

See zeta constants.

infty
\zeta(2n)=\sum
k=1
1
k2n

=(-1)n+1

B2n(2\pi)2n
2(2n)!

The first few values are:
infty
\zeta(2)=\sum
k=1
1=
k2
\pi2
6
(the Basel problem)
infty
\zeta(4)=\sum
k=1
1=
k4
\pi4
90
infty
\zeta(6)=\sum
k=1
1=
k6
\pi6
945

Power series

Low-order polylogarithms

Finite sums:

n
\sum
k=m

zk=

zm-zn+1
1-z
, (geometric series)
n
\sum
k=0

zk=

1-zn+1
1-z
n
\sum
k=1

zk=

1-zn+1
1-z

-1=

z-zn+1
1-z
n
\sum
k=1

kzk=z

1-(n+1)zn+nzn+1
(1-z)2
n
\sum
k=1

k2zk=z

1+z-(n+1)2zn+(2n2+2n-1)zn+1-n2zn+2
(1-z)3

n
\sum
k=1

kmzk=\left(z

d
dz

\right)m

1-zn+1
1-z

Infinite sums, valid for

|z|<1

(see polylogarithm):

\operatorname{Li}n(z)=\sum

infty
k=1
zk
kn
The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:
d
dz
\operatorname{Li}
n(z)=\operatorname{Li
n-1

(z)}{z}

\operatorname{Li}1

infty
(z)=\sum
k=1
zk
k

=-ln(1-z)

\operatorname{Li}0

infty
(z)=\sum
k=1
k=z
1-z
z

\operatorname{Li}-1

infty
(z)=\sum
k=1

k

k=z
(1-z)2
z

\operatorname{Li}-2

infty
(z)=\sum
k=1

k2

k=z(1+z)
(1-z)3
z

\operatorname{Li}-3

infty
(z)=\sum
k=1

k3zk=

z(1+4z+z2)
(1-z)4

\operatorname{Li}-4

infty
(z)=\sum
k=1

k4zk=

z(1+z)(1+10z+z2)
(1-z)5

Exponential function

infty
\sum
k=0
zk
k!

=ez

infty
\sumk
k=0
zk
k!

=zez

(cf. mean of Poisson distribution)
infty
\sum
k=0

k2

zk
k!

=(z+z2)ez

(cf. second moment of Poisson distribution)
infty
\sum
k=0

k3

zk
k!

=(z+3z2+z3)ez

infty
\sum
k=0

k4

zk
k!

=(z+7z2+6z3+z4)ez

infty
\sum
k=0

kn

zk
k!

=z

d
dz
infty
\sum
k=0

kn-1

zk
k!

=ezTn(z)

where

Tn(z)

is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship

infty
\sum
k=0
(-1)kz2k+1
(2k+1)!

=\sinz

infty
\sum
k=0
z2k+1
(2k+1)!

=\sinhz

infty
\sum
k=0
(-1)kz2k
(2k)!

=\cosz

infty
\sum
k=0
z2k
(2k)!

=\coshz

infty
\sum
k=1
(-1)k-1(22k-1)22kB2kz2k-1
(2k)!

=\tanz,|z|<

\pi
2
infty
\sum
k=1
(22k-1)22kB2kz2k-1
(2k)!

=\tanhz,|z|<

\pi
2
infty
\sum
k=0
(-1)k22kB2kz2k-1
(2k)!

=\cotz,|z|<\pi

infty
\sum
k=0
22kB2kz2k-1
(2k)!

=\cothz,|z|<\pi

infty
\sum
k=0
(-1)k-1(22k-2)B2kz2k-1
(2k)!

=\cscz,|z|<\pi

infty
\sum
k=0
-(22k-2)B2kz2k-1
(2k)!

=\operatorname{csch}z,|z|<\pi

infty
\sum
k=0
kE
(-1)z2k
2k
(2k)!

=\operatorname{sech}z,|z|<

\pi
2
infty
\sum
k=0
E2kz2k
(2k)!

=\secz,|z|<

\pi
2
infty
\sum
k=1
(-1)k-1z2k
(2k)!

=\operatorname{ver}z

(versine)
infty
\sum
k=1
(-1)k-1z2k
2(2k)!

=\operatorname{hav}z

[1] (haversine)
infty
\sum
k=0
(2k)!z2k+1
22k(k!)2(2k+1)

=\arcsinz,|z|\le1

infty
\sum
k=0
(-1)k(2k)!z2k+1
22k(k!)2(2k+1)

=\operatorname{arcsinh}{z},|z|\le1

infty
\sum
k=0
(-1)kz2k+1
2k+1

=\arctanz,|z|<1

infty
\sum
k=0
z2k+1
2k+1

=\operatorname{arctanh}z,|z|<1

infty
ln2+\sum
k=1
(-1)k-1(2k)!z2k
22k+1k(k!)2

=ln\left(1+\sqrt{1+z2}\right),|z|\le1

infty
\sum
k=2

\left(k\operatorname{arctanh}\left(

1
k

\right)-1\right)=

3-ln(4\pi)
2

Modified-factorial denominators

infty
\sum
k=0
(4k)!
24k\sqrt{2

(2k)!(2k+1)!}zk=\sqrt{

1-\sqrt{1-z
}}, |z|<1[2]
infty
\sum
k=0
22k(k!)2
(k+1)(2k+1)!

z2k+2=\left(\arcsin{z}\right)2,|z|\le1

[2]
infty
\sum
n=0
n-1
\prod(4k2+\alpha2)
k=0
(2n)!

z2n+

infty
\sum
n=0
\alpha
n-1
\prod
k=0
[(2k+1)2+\alpha2]
(2n+1)!

z2n+1=e\alpha

}, |z|\le1

Binomial coefficients

(1+z)\alpha=

infty
\sum
k=0

{\alpha\choosek}zk,|z|<1

(see)
infty
\sum
k=0

{{\alpha+k-1}\choosek}zk=

1
(1-z)\alpha

,|z|<1

infty
\sum
k=0
1
k+1

{2k\choosek}zk=

1-\sqrt{1-4z
}, |z|\leq\frac, generating function of the Catalan numbers
infty
\sum
k=0

{2k\choosek}zk=

1
\sqrt{1-4z
}, |z|<\frac, generating function of the Central binomial coefficients
infty
\sum
k=0

{2k+\alpha\choosek}zk=

1
\sqrt{1-4z
}\left(\frac\right)^\alpha, |z|<\frac

Harmonic numbers

(See harmonic numbers, themselves defined H_n = \sum_^ \frac , and

H(x)

generalized to the real numbers)
infty
\sum
k=1

Hkzk=

-ln(1-z)
1-z

,|z|<1

infty
\sum
k=1
Hk
k+1

zk+1=

1
2

\left[ln(1-z)\right]2,    |z|<1

infty
\sum
k=1
(-1)k-1H2k
2k+1

z2k+1=

1
2

\arctan{z}log{(1+z2)},    |z|<1

[2]
infty
\sum
n=0
2n
\sum
k=0
(-1)k
2k+1
z4n+2
4n+2

=

1
4

\arctan{z}log{

1+z
1-z
},\qquad |z|<1 [2]
infty
\sum
n=0
x2
n2(n+x)

=x

\pi2
6

-H(x)

Binomial coefficients

See main article: Binomial coefficient.

n
\sum
k=0

{n\choosek}=2n

n
\sum
k=0

{n\choosek}2={2n\choosen}

n
\sum
k=0

(-1)k{n\choosek}=0,wheren\geq1

n
\sum
k=0

{k\choosem}={n+1\choosem+1}

n
\sum
k=0

{m+k-1\choosek}={n+m\choosen}

(see Multiset)
n
\sum
k=0

{\alpha\choosek}{\beta\choosen-k}={\alpha+\beta\choosen},where\alpha+\beta\geqn

(see Vandermonde identity)

\sumA(E)}1=2n,whereEisafiniteset,andcard(E)=n

\sum\begin{cases(A,B)\in(l{P}(E))2\A\subsetB\end{cases}}1=3n,whereEisafiniteset,andcard(E)=n

\sumA(E)}card(A)=n2n-1,whereEisafiniteset,andcard(E)=n

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

infty
\sum
k=1
\cos(k\theta)=-
k
1
2

ln(2-2\cos\theta)=-ln\left(2\sin

\theta
2

\right),0<\theta<2\pi

infty
\sum
k=1
\sin(k\theta)=
k
\pi-\theta
2

,0<\theta<2\pi

infty
\sum
k=1
(-1)k-1\cos(k\theta)=
k
1
2

ln(2+2\cos\theta)=ln\left(2\cos

\theta
2

\right),0\leq\theta<\pi

infty
\sum
k=1
(-1)k-1\sin(k\theta)=
k
\theta
2

,-

\pi\leq\theta\leq
2
\pi
2
infty
\sum
k=1
\cos(2k\theta)=-
2k
1
2

ln(2\sin\theta),0<\theta<\pi

infty
\sum
k=1
\sin(2k\theta)=
2k
\pi-2\theta
4

,0<\theta<\pi

infty
\sum
k=0
\cos[(2k+1)\theta]=
2k+1
1
2

ln\left(\cot

\theta
2

\right),0<\theta<\pi

infty
\sum
k=0
\sin[(2k+1)\theta]=
2k+1
\pi
4

,0<\theta<\pi

,[4]
infty
\sum
k=1
\sin(2\pikx)
k

=\pi\left(\dfrac{1}{2}-\{x\}\right),x\inR

infty
\sum\limits
k=1
\sin\left(2\pikx\right)
k2n-1

=(-1)n

(2\pi)2n-1
2(2n-1)!

B2n-1(\{x\}),x\inR,n\inN

infty
\sum\limits
k=1
\cos\left(2\pikx\right)
k2n

=(-1)n-1

(2\pi)2n
2(2n)!

B2n(\{x\}),x\inR,n\inN

B
n(x)=-n!
2n-1\pin
infty
\sum
k=1
1
kn

\cos\left(2\pikx-

\pin
2

\right),0<x<1

[5]
n
\sum\sin(\theta+k\alpha)=
k=0
\sin(n+1)\alpha
\sin(\theta+n\alpha
2
)
2
\sin\alpha
2
n
\sum\cos(\theta+k\alpha)=
k=0
\sin(n+1)\alpha
\cos(\theta+n\alpha
2
)
2
\sin\alpha
2
n-1
\sum\sin
k=1
\pik=\cot
n
\pi
2n
n-1
\sum\sin
k=1
2\pik
n

=0

n-1
\sum
k=0
2\left(\theta+\pik
n
\csc

\right)=n2\csc2(n\theta)

[6]
n-1
\sum
k=1
2\pik
n
\csc=
n2-1
3
n-1
\sum
k=1
4\pik
n
\csc=
n4+10n2-11
45

Rational functions

infty
\sum
n=a+1
a
n2-a2

=

1
2

H2a

[7]
infty1
n2+a2
\sum=
n=0
1+a\pi\coth(a\pi)
2a2
infty(-1)n
n2+a2
\sum
n=0

=

1+a\picsch(a\pi)
2a2
infty(2n+1)(-1)n
(2n+1)2+a2
\sum
n=0

=

\pi
4

sech\left(

a\pi
2

\right)

\displaystyle

infty
\sum
n=0
1
n4+4a4

=\dfrac{1}{8a4}+\dfrac{\pi(\sinh(2\pia)+\sin(2\pia))}{8a3(\cosh(2\pia)-\cos(2\pia))}

n

can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition,[8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.

Exponential function

\displaystyle

p-1
\dfrac{1}{\sqrt{p}}\sum
n=0

\exp\left(

2\piin2q
p

\right)=\dfrac{e\pi

}\sum_^\exp \left(-\frac \right)(see the Landsberg–Schaar relation)

\displaystyle

infty
\sum
n=-infty
-\pin2
e

=

\sqrt[4]\pi
\Gamma\left(3
4\right)

Numeric series

These numeric series can be found by plugging in numbers from the series listed above.

Alternating harmonic series

infty
\sum
k=1
(-1)k+1=
k
1-
1
1+
2
1-
3
1
4

+ … =ln2

infty
\sum
k=1
(-1)k+1=
2k-1
1-
1
1+
3
1-
5
1+
7
1- … =
9
\pi
4

Sum of reciprocal of factorials

infty
\sum
k=0
1=
k!
1+
0!
1+
1!
1+
2!
1+
3!
1
4!

+ … =e

infty
\sum
k=0
1=
(2k)!
1+
0!
1+
2!
1+
4!
1+
6!
1+ … =
8!
1\left(e+
2
1
e

\right)=\cosh1

infty
\sum
k=0
1=
(3k)!
1+
0!
1+
3!
1+
6!
1+
9!
1+ … =
12!
1\left(e+
3
2
\sqrt{e
}\cos \frac\right)
infty
\sum
k=0
1=
(4k)!
1+
0!
1+
4!
1+
8!
1+
12!
1+ … =
16!
1
2

\left(\cos1+\cosh1\right)

Trigonometry and π

infty
\sum
k=0
(-1)k=
(2k+1)!
1-
1!
1+
3!
1-
5!
1+
7!
1
9!

+ … =\sin1

infty
\sum
k=0
(-1)k=
(2k)!
1-
0!
1+
2!
1-
4!
1+
6!
1
8!

+ … =\cos1

infty
\sum
k=1
1=
k2+1
1+
2
1+
5
1+
10
1+ … =
17
1
2

(\pi\coth\pi-1)

infty
\sum
k=1
(-1)k=-
k2+1
1+
2
1-
5
1+
10
1+ … =
17
1
2

(\pi\operatorname{csch}\pi-1)

3+

4
2 x 3 x 4

-

4
4 x 5 x 6

+

4
6 x 7 x 8

-

4
8 x 9 x 10

+=\pi

Reciprocal of tetrahedral numbers

infty
\sum
k=1
1=
Tek
1+
1
1+
4
1+
10
1+
20
1+ … =
35
3
2
Where
n
Te
k=1

Tk

Exponential and logarithms

infty
\sum
k=0
1=
(2k+1)(2k+2)
1+
1 x 2
1+
3 x 4
1+
5 x 6
1+
7 x 8
1
9 x 10

+ … =ln2

infty
\sum
k=1
1=
2kk
1+
2
1+
8
1+
24
1+
64
1
160

+ … =ln2

infty
\sum
k=1
(-1)k+1
2kk
infty
+\sum
k=1
(-1)k+1=(
3kk
1+
2
1)-(
3
1+
8
1)+(
18
1+
24
1)-(
81
1+
64
1
324

)+ … =ln2

infty
\sum
k=1
1
3kk
infty
+\sum
k=1
1=(
4kk
1+
3
1)+(
4
1+
18
1)+(
32
1+
81
1)+(
192
1+
324
1
1024

)+ … =ln2

infty
\sum
k=1
1=ln\left(
nkk
n
n-1

\right)

, that is

\foralln>1

See also

References

Notes and References

  1. Web site: Eric W. . Weisstein . Eric W. Weisstein . Haversine . . . 2015-11-06 . live . https://web.archive.org/web/20050310194740/http://mathworld.wolfram.com/Haversine.html . 2005-03-10.
  2. Book: Wilf, Herbert R.. generatingfunctionology. 1994. Academic Press, Inc.
  3. Web site: Theoretical computer science cheat sheet.
  4. Calculate the Fourier expansion of the function
    f(x)=\pi4
    on the interval

    0<x<\pi

    :
    \pi4=\sum
    n=0

    inftycn\sin[nx]+dn\cos[nx]

    \begin{cases}c
    n=\begin{cases}1n
    (n

    odd)\\ 0   (neven)\end{cases}\dn=0   (\foralln)\end{cases}

  5. Web site: Bernoulli polynomials: Series representations (subsection 06/02). 2 June 2011. Wolfram Research.
  6. Web site: Hofbauer. Josef. A simple proof of 1 + 1/22 + 1/32 + &middot;&middot;&middot; = 2/6 and related identities. 2 June 2011.
  7. Web site: Sondow. Jonathan. Weisstein. Eric W.. Riemann Zeta Function (eq. 52). MathWorld—A Wolfram Web Resource.
  8. Book: http://people.math.sfu.ca/~cbm/aands/page_260.htm. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Milton. Abramowitz. Milton Abramowitz. Irene. Stegun. Irene Stegun. 1964. 0-486-61272-4. 260. 6.4 Polygamma functions. Courier Corporation .