finite sum

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21: 29.20 Methods of Computation

… ►The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … ►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices $𝐌$ given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …

22: 1.9 Calculus of a Complex Variable

… ►

1.9.68 $${\int }_C\sum _{n=0}^{\mathrm{\infty }}{f}_n(z)dz=\sum _{n=0}^{\mathrm{\infty }}{\int }_C{f}_n(z)dz$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: variable, $n$: nonnegative integer and $C$: finite contour in $D$

Permalink:

http://dlmf.nist.gov/1.9.E68

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Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

…

23: 6.6 Power Series

… ►

6.6.1 $$\mathrm{Ei}\left(x\right)=\gamma +\ln x+\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n!n},$$ $x>0$.

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

5.1.10

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E1

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Annotations for §6.6 and Ch.6

►

6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

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Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.1.11

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.6.E2

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Annotations for §6.6 and Ch.6

… ►

6.6.4 $$\mathrm{Ein}\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{z}^n}{n!n},$$

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.6.E4

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Annotations for §6.6 and Ch.6

►

6.6.5 $$\mathrm{Si}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{(2n+1)!(2n+1)},$$

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{Si}\left(z\right)$: sine integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.14

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E5

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Annotations for §6.6 and Ch.6

… ►The series in this section converge for all finite values of $x$ and $|z|$.

24: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. Cancellation errors increase with increases in $\rho$ and $|r|$, and may be estimated by comparing the final sum of the series with the largest partial sum. …

25: 24.4 Basic Properties

… ►

§24.4(iii) Sums of Powers

►

24.4.7 $$\sum _{k=1}^m{k}^n=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E7

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Annotations for §24.4(iii), §24.4 and Ch.24

►

24.4.8 $$\sum _{k=1}^m{(-1)}^{m-k}{k}^n=\frac{E_n\left(m+1\right)+{(-1)}^m{E}_n\left(0\right)}{2}.$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E8

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Annotations for §24.4(iii), §24.4 and Ch.24

… ►

§24.4(iv) Finite Expansions

… ►

24.4.24 $$B_n\left(mx\right)=m^n{B}_n\left(x\right)+n\sum _{k=1}^n\sum _{j=0}^{k-1}{(-1)}^j\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\sum _{r=1}^{m-1}\frac{{\mathrm{e}}^{2\pi \mathrm{i}(k-j)r/m}}{{(1-{\mathrm{e}}^{2\pi \mathrm{i}r/m})}^n}\right){(j+mx)}^{n-1},$$ $n=1,2,\mathrm{\dots }$, $m=2,3,\mathrm{\dots }$.

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $j$: integer, $k$: integer, $m$: integer, $n$: integer and $x$: real or complex

Referenced by:

§24.4(v)

Permalink:

http://dlmf.nist.gov/24.4.E24

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See also:

Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

…

26: 1.12 Continued Fractions

… ►

1.12.18 $$p_0+\sum _{k=1}^n{p}_1{p}_2\mathrm{\cdots }{p}_k=p_0+\frac{p_1}{1-}\frac{p_2}{1+p_2-}\frac{p_3}{1+p_3-}\mathrm{\cdots }{\displaystyle \frac{p_n}{1+p_n}},$$ $n=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$k$: integer, $n$: nonnegative integer and $p_k$: coefficients

Permalink:

http://dlmf.nist.gov/1.12.E18

Encodings:

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See also:

Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

… ►

1.12.19 $$\sum _{k=0}^n{c}_k{x}^k=c_0+\frac{c_{1}x}{1-}\frac{(c_2/c_1)x}{1+(c_2/c_1)x-}\frac{(c_3/c_2)x}{1+(c_3/c_2)x-}\mathrm{\cdots }{\displaystyle \frac{(c_n/c_{n-1})x}{1+(c_n/c_{n-1})x}},$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$k$: integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.12.E19

Encodings:

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Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

… ►A continued fraction converges if the convergents $C_n$ tend to a finite limit as $n\to \mathrm{\infty }$. … ►and the even and odd parts of the continued fraction converge to finite values. … ►

1.12.28 $$\sum _{n=1}^{\mathrm{\infty }}\left|b_n\right|=\mathrm{\infty }.$$

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Symbols:

$n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.12.E28

Encodings:

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Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

…

27: Bibliography J

… ►

Note sur la série ${\sum }_{n=1}^{\mathrm{\infty }}{x}^n/n^s$ . Bull. Soc. Math. France 17, pp. 142–152 (French).

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External Links:

ISSN 0037-9484, MathReview Entry, ZentralBlatt

Referenced by:

§25.12(ii).

Encodings:

BibTeX

►

Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

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Notes:

Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI

External Links:

MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§26.1, §26.1, §26.8(vii), §5.16, Notations S.

Encodings:

BibTeX

►

Calculus of Finite Differences. 3rd edition, AMS Chelsea, Providence, RI.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§24.17(i), §24.6.

Encodings:

BibTeX

…

28: 2.3 Integrals of a Real Variable

… ►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$. …

29: 26.18 Counting Techniques

… ►

26.18.1

ⓘ

Symbols:

$\left|A\right|$: number of elements of a finite set $A$ , $\cap$: intersection, $\setminus$: set subtraction, $\cup$: union, $j$: nonnegative integer, $n$: nonnegative integer, $A_k$: subsets and $S$: set

Permalink:

http://dlmf.nist.gov/26.18.E1

Encodings:

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See also:

Annotations for §26.18 and Ch.26

…

30: 7.6 Series Expansions

… ►

7.6.1 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{n!(2n+1)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.5

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E1

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Annotations for §7.6(i), §7.6 and Ch.7

►

7.6.2 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}\sum _{n=0}^{\mathrm{\infty }}\frac{2^n{z}^{2n+1}}{1\cdot 3\mathrm{\cdots }(2n+1)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.6

Referenced by:

§7.6(i), §7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E2

Encodings:

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Annotations for §7.6(i), §7.6 and Ch.7

… ►

7.6.5 $$C\left(z\right)=\cos \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n}}{1\cdot 3\mathrm{\cdots }(4n+1)}{z}^{4n+1}+\sin \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n+1}}{1\cdot 3\mathrm{\cdots }(4n+3)}{z}^{4n+3}.$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.12

Referenced by:

§11.10(vi), §7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E5

Encodings:

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Annotations for §7.6(i), §7.6 and Ch.7

… ►

7.6.7 $$S\left(z\right)=-\cos \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n+1}}{1\cdot 3\mathrm{\cdots }(4n+3)}{z}^{4n+3}+\sin \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n}}{1\cdot 3\mathrm{\cdots }(4n+1)}{z}^{4n+1}.$$

ⓘ

Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.14

Referenced by:

§11.10(vi), §7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E7

Encodings:

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See also:

Annotations for §7.6(i), §7.6 and Ch.7

►The series in this subsection and in §7.6(ii) converge for all finite values of $|z|$. …