§1.15 Summability Methods

§1.15(i) Definitions for Series

 1.15.1 $s_{n}=\sum_{k=0}^{n}a_{k}.$ ⓘ Symbols: $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E1 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15 and Ch.1

Abel Summability

 1.15.2 $\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(A)},$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E2 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

if

 1.15.3 $\lim_{x\to 1-}\sum^{\infty}_{n=0}a_{n}x^{n}=s.$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E3 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

Cesàro Summability

 1.15.4 $\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(C,1)},$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E4 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

if

 1.15.5 $\lim_{n\to\infty}\frac{s_{0}+s_{1}+\dots+s_{n}}{n+1}=s.$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E5 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

General Cesàro Summability

For $\alpha>-1$,

 1.15.6 $\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(C,\alpha)},$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E6 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

if

 1.15.7 $\lim_{n\to\infty}\frac{n!}{(\alpha+1)_{n}}\sum^{n}_{k=0}\frac{(\alpha+1)_{k}}{% k!}a_{n-k}=s.$ ⓘ Symbols: $!$: factorial (as in $n!$), $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E7 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

Borel Summability

 1.15.8 $\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(B)},$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E8 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

if

 1.15.9 $\lim_{t\to\infty}e^{-t}\sum^{\infty}_{n=0}\frac{s_{n}}{n!}t^{n}=s.$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$) and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E9 Encodings: TeX, pMML, png See also: Annotations for §1.15(i), §1.15(i), §1.15 and Ch.1

§1.15(ii) Regularity

Methods of summation are regular if they are consistent with conventional summation. All of the methods described in §1.15(i) are regular. For example if

 1.15.10 $\sum^{\infty}_{n=0}a_{n}=s,$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E10 Encodings: TeX, pMML, png See also: Annotations for §1.15(ii), §1.15 and Ch.1

then

 1.15.11 $\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(A)}.$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E11 Encodings: TeX, pMML, png See also: Annotations for §1.15(ii), §1.15 and Ch.1

§1.15(iii) Summability of Fourier Series

Poisson Kernel

 1.15.12 $P(r,\theta)=\frac{1-r^{2}}{1-2r\cos\theta+r^{2}}=\sum^{\infty}_{n=-\infty}r^{|% n|}e^{in\theta},$ $0\leq r<1$, ⓘ Defines: $P(r,\theta)$: Poisson kernel (locally) Symbols: $\cos\NVar{z}$: cosine function, $\mathrm{e}$: base of natural logarithm and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E12 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1
 1.15.13 $\frac{1}{2\pi}\int^{2\pi}_{0}P(r,\theta)\mathrm{d}\theta=1.$

As $r\to 1-$

 1.15.14 $P(r,\theta)\to 0,$ ⓘ Symbols: $P(r,\theta)$: Poisson kernel Permalink: http://dlmf.nist.gov/1.15.E14 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

uniformly for $\theta\in[\delta,2\pi-\delta]$. (Here and elsewhere in this subsection $\delta$ is a constant such that $0<\delta<\pi$.)

Fejér Kernel

For $n=0,1,2,\dots$,

 1.15.15 $K_{n}(\theta)=\frac{1}{n+1}\left(\frac{\sin\left(\tfrac{1}{2}(n+1)\theta\right% )}{\sin\left(\tfrac{1}{2}\theta\right)}\right)^{2},$ ⓘ Defines: $K_{n}(\theta)$: Fejér kernel (locally) Symbols: $\sin\NVar{z}$: sine function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E15 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1
 1.15.16 $\frac{1}{2\pi}\int^{2\pi}_{0}K_{n}(\theta)\mathrm{d}\theta=1.$

As $n\to\infty$

 1.15.17 $K_{n}(\theta)\to 0,$ ⓘ Symbols: $n$: nonnegative integer and $K_{n}(\theta)$: Fejér kernel Permalink: http://dlmf.nist.gov/1.15.E17 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

uniformly for $\theta\in[\delta,2\pi-\delta]$.

Abel Means

 1.15.18 $A(r,\theta)=\sum^{\infty}_{n=-\infty}r^{|n|}F(n)e^{in\theta},$ ⓘ Defines: $A(r,\theta)$: Abel mean (locally) Symbols: $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $F(n)$ Permalink: http://dlmf.nist.gov/1.15.E18 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

where

 1.15.19 $F(n)=\frac{1}{2\pi}\int^{2\pi}_{0}f(t)e^{-int}\mathrm{d}t.$ ⓘ Defines: $F(n)$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E19 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

$A(r,\theta)$ is a harmonic function in polar coordinates ((1.9.27)), and

 1.15.20 $A(r,\theta)=\frac{1}{2\pi}\int^{2\pi}_{0}P(r,\theta-t)f(t)\mathrm{d}t.$

Cesàro (or (C,1)) Means

Let

 1.15.21 $\sigma_{n}(\theta)=\frac{s_{0}(\theta)+s_{1}(\theta)+\dots+s_{n}(\theta)}{n+1},$ ⓘ Defines: $\sigma_{n}(\theta)$: Cesàro mean (locally) Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E21 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

$n=0,1,2,\dots$, where

 1.15.22 $s_{n}(\theta)=\sum^{n}_{k=-n}F(k)e^{ik\theta}.$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $k$: integer, $n$: nonnegative integer and $F(n)$ Permalink: http://dlmf.nist.gov/1.15.E22 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

Then

 1.15.23 $\sigma_{n}(\theta)=\frac{1}{2\pi}\int^{2\pi}_{0}K_{n}(\theta-t)f(t)\mathrm{d}t.$

Convergence

If $f(\theta)$ is periodic and integrable on $[0,2\pi]$, then as $n\to\infty$ the Abel means $A(r,\theta)$ and the (C,1) means $\sigma_{n}(\theta)$ converge to

 1.15.24 $\tfrac{1}{2}(f(\theta+)+f(\theta-))$ ⓘ Referenced by: §1.15(iii) Permalink: http://dlmf.nist.gov/1.15.E24 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

at every point $\theta$ where both limits exist. If $f(\theta)$ is also continuous, then the convergence is uniform for all $\theta$.

For real-valued $f(\theta)$, if

 1.15.25 $\sum^{\infty}_{n=-\infty}F(n)e^{in\theta}$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $F(n)$ Permalink: http://dlmf.nist.gov/1.15.E25 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

is the Fourier series of $f(\theta)$, then the series

 1.15.26 $F(0)+2\sum^{\infty}_{n=1}F(n)e^{in\theta}$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $F(n)$ Permalink: http://dlmf.nist.gov/1.15.E26 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

can be extended to the interior of the unit circle as an analytic function

 1.15.27 $G(z)=G(x+iy)=u(x,y)+iv(x,y)=F(0)+2\sum^{\infty}_{n=1}F(n)z^{n}.$ ⓘ Symbols: $(\NVar{a},\NVar{b})$: open interval, $z$: variable, $n$: nonnegative integer, $F(n)$ and $u(x,y)$: function Permalink: http://dlmf.nist.gov/1.15.E27 Encodings: TeX, pMML, png See also: Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

Here $u(x,y)=A(r,\theta)$ is the Abel (or Poisson) sum of $f(\theta)$, and $v(x,y)$ has the series representation

 1.15.28 $-\sum^{\infty}_{n=-\infty}i(\operatorname{sign}n)F(n)r^{|n|}e^{in\theta};$

compare §1.15(v).

§1.15(iv) Definitions for Integrals

Abel Summability

$\int^{\infty}_{-\infty}f(t)\mathrm{d}t$ is Abel summable to $L$, or

 1.15.29 $\int^{\infty}_{-\infty}f(t)\mathrm{d}t=L\;\;\;\textit{(A)},$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.15.E29 Encodings: TeX, pMML, png See also: Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

when

 1.15.30 $\lim_{\epsilon\to 0+}\int^{\infty}_{-\infty}e^{-\epsilon|t|}f(t)\mathrm{d}t=L.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm and $\int$: integral Permalink: http://dlmf.nist.gov/1.15.E30 Encodings: TeX, pMML, png See also: Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

Cesàro Summability

$\int^{\infty}_{-\infty}f(t)\mathrm{d}t$ is (C,1) summable to $L$, or

 1.15.31 $\int^{\infty}_{-\infty}f(t)\mathrm{d}t=L\;\;\;\textit{(C,1)},$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.15.E31 Encodings: TeX, pMML, png See also: Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

when

 1.15.32 $\lim_{R\to\infty}\int^{R}_{-R}\left(1-\frac{|t|}{R}\right)f(t)\mathrm{d}t=L.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.15.E32 Encodings: TeX, pMML, png See also: Annotations for §1.15(iv), §1.15(iv), §1.15 and Ch.1

If $\int^{\infty}_{-\infty}f(t)\mathrm{d}t$ converges and equals $L$, then the integral is Abel and Cesàro summable to $L$.

§1.15(v) Summability of Fourier Integrals

Poisson Kernel

 1.15.33 $P(x,y)=\frac{2y}{x^{2}+y^{2}},$ $y>0$, $-\infty. ⓘ Defines: $P(x,y)$: Poisson kernel (locally) Permalink: http://dlmf.nist.gov/1.15.E33 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1
 1.15.34 $\frac{1}{2\pi}\int^{\infty}_{-\infty}P(x,y)\mathrm{d}x=1.$

For each $\delta>0$,

 1.15.35 $\int_{|x|\geq\delta}P(x,y)\mathrm{d}x\to 0,$ as $y\to 0$. ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $P(x,y)$: Poisson kernel Permalink: http://dlmf.nist.gov/1.15.E35 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

Let

 1.15.36 $h(x,y)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}e^{-y|t|}e^{-ixt}F(t)% \mathrm{d}t,$

where $F(t)$ is the Fourier transform of $f(x)$1.14(i)). Then

 1.15.37 $h(x,y)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(t)P(x-t,y)\mathrm{d}t$

is the Poisson integral of $f(t)$.

If $f(x)$ is integrable on $(-\infty,\infty)$, then

 1.15.38 $\lim_{y\to 0+}\int^{\infty}_{-\infty}|h(x,y)-f(x)|\mathrm{d}x=0.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $h(x,y)$: function Permalink: http://dlmf.nist.gov/1.15.E38 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

Suppose now $f(x)$ is real-valued and integrable on $(-\infty,\infty)$. Let

 1.15.39 $\Phi(z)=\Phi(x+iy)=\frac{i}{\pi}\int^{\infty}_{-\infty}f(t)\frac{1}{(x-t)+iy}% \mathrm{d}t,$

where $y>0$ and $-\infty. Then $\Phi(z)$ is an analytic function in the upper half-plane and its real part is the Poisson integral $h(x,y)$; compare (1.9.34). The imaginary part

 1.15.40 $\Im\Phi(x+iy)=\frac{1}{\pi}\int^{\infty}_{-\infty}f(t)\frac{x-t}{(x-t)^{2}+y^{% 2}}\mathrm{d}t$

is the conjugate Poisson integral of $f(x)$. Moreover, $\lim_{y\to 0+}\Im\Phi(x+iy)$ is the Hilbert transform of $f(x)$1.14(v)).

Fejér Kernel

 1.15.41 $K_{R}(s)=\frac{1}{\pi R}\frac{1-\cos\left(Rs\right)}{s^{2}},$ ⓘ Defines: $K_{R}(s)$: Fejér kernel (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\cos\NVar{z}$: cosine function Permalink: http://dlmf.nist.gov/1.15.E41 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1
 1.15.42 $\int^{\infty}_{-\infty}K_{R}(s)\mathrm{d}s=1.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $K_{R}(s)$: Fejér kernel Permalink: http://dlmf.nist.gov/1.15.E42 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

For each $\delta>0$,

 1.15.43 $\int_{|s|\geq\delta}K_{R}(s)\mathrm{d}s\to 0,$ as $R\to\infty$. ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $K_{R}(s)$: Fejér kernel Permalink: http://dlmf.nist.gov/1.15.E43 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

Let

 1.15.44 $\displaystyle\sigma_{R}(\theta)$ $\displaystyle=\frac{1}{\sqrt{2\pi}}\int^{R}_{-R}\left(1-\frac{|t|}{R}\right)e^% {-i\theta t}F(t)\mathrm{d}t,$ then 1.15.45 $\displaystyle\sigma_{R}(\theta)$ $\displaystyle=\int^{\infty}_{-\infty}f(t)K_{R}(\theta-t)\mathrm{d}t.$

If $f(\theta)$ is integrable on $(-\infty,\infty)$, then

 1.15.46 $\lim_{R\to\infty}\int^{\infty}_{-\infty}|\sigma_{R}(\theta)-f(\theta)|\mathrm{% d}\theta=0.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\sigma_{R}(\theta)$: function Permalink: http://dlmf.nist.gov/1.15.E46 Encodings: TeX, pMML, png See also: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

§1.15(vi) Fractional Integrals

For $\Re\alpha>0$ and $x\geq 0$, the Riemann-Liouville fractional integral of order $\alpha$ is defined by

 1.15.47 $I^{\alpha}f(x)=\frac{1}{\Gamma\left(\alpha\right)}\int^{x}_{0}(x-t)^{\alpha-1}% f(t)\mathrm{d}t.$ ⓘ Defines: $I^{\alpha}$: fractional integral (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §1.15(vi), §1.15(vi), 1st Erratum Permalink: http://dlmf.nist.gov/1.15.E47 Encodings: TeX, pMML, png See also: Annotations for §1.15(vi), §1.15 and Ch.1

For $\Gamma\left(\alpha\right)$ see §5.2, and compare (1.4.31) in the case when $\alpha$ is a positive integer.

 1.15.48 $I^{\alpha}I^{\beta}=I^{\alpha+\beta},$ $\Re\alpha>0$, $\Re\beta>0$. ⓘ Symbols: $\Re$: real part and $I^{\alpha}$: fractional integral Referenced by: §1.15(vi) Permalink: http://dlmf.nist.gov/1.15.E48 Encodings: TeX, pMML, png See also: Annotations for §1.15(vi), §1.15 and Ch.1

For extensions of (1.15.48) see Love (1972b).

If

 1.15.49 $f(x)=\sum^{\infty}_{k=0}a_{k}x^{k},$ ⓘ Symbols: $k$: integer Permalink: http://dlmf.nist.gov/1.15.E49 Encodings: TeX, pMML, png See also: Annotations for §1.15(vi), §1.15 and Ch.1

then

 1.15.50 $I^{\alpha}f(x)=\sum^{\infty}_{k=0}\frac{k!}{\Gamma\left(k+\alpha+1\right)}a_{k% }x^{k+\alpha}.$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $!$: factorial (as in $n!$), $k$: integer and $I^{\alpha}$: fractional integral Referenced by: 1st Erratum Permalink: http://dlmf.nist.gov/1.15.E50 Encodings: TeX, pMML, png See also: Annotations for §1.15(vi), §1.15 and Ch.1

The lower limit $0$ of the integral in (1.15.47) can be replaced by any constant $a\leq x$ . Also, we can replace the lower and upper limits of the integral by $x$ and $a$, respectively. In that case we must also replace $(x-t)$ in the integrand by $(t-x)$ and we can even set $a=\infty$. See (18.17.9), (18.17.11) and (18.17.13) as examples.

§1.15(vii) Fractional Derivatives

For $0<\Re\alpha, $n$ an integer, and $x\geq 0$, the fractional derivative of order $\alpha$ is defined by

 1.15.51 $D^{\alpha}f(x)=\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}I^{n-\alpha}f(x),$ ⓘ Defines: $D^{\alpha}$: fractional derivative (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $n$: nonnegative integer and $I^{\alpha}$: fractional integral Referenced by: §1.15(vii) Permalink: http://dlmf.nist.gov/1.15.E51 Encodings: TeX, pMML, png See also: Annotations for §1.15(vii), §1.15 and Ch.1

and satisfies the property

 1.15.52 $D^{k}I^{\alpha}=D^{n}I^{\alpha+n-k},$ $k=1,2,\dots,n$. ⓘ Symbols: $k$: integer, $n$: nonnegative integer, $I^{\alpha}$: fractional integral and $D^{\alpha}$: fractional derivative Referenced by: §1.15(vii) Permalink: http://dlmf.nist.gov/1.15.E52 Encodings: TeX, pMML, png See also: Annotations for §1.15(vii), §1.15 and Ch.1

When none of $\alpha$, $\beta$, and $\alpha+\beta$ is an integer

 1.15.53 $D^{\alpha}D^{\beta}=D^{\alpha+\beta}.$ ⓘ Symbols: $D^{\alpha}$: fractional derivative Permalink: http://dlmf.nist.gov/1.15.E53 Encodings: TeX, pMML, png See also: Annotations for §1.15(vii), §1.15 and Ch.1

Note that $D^{1/2}D\not=D^{3/2}$. See also Love (1972b).

§1.15(viii) Tauberian Theorems

If

 1.15.54 $\displaystyle\sum^{\infty}_{n=0}a_{n}$ $\displaystyle=s\;\;\;\textit{(A)},$ $\displaystyle a_{n}$ $\displaystyle>-\frac{K}{n}$, $n>0$, $K>0$, ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E54 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.15(viii), §1.15 and Ch.1

then

 1.15.55 $\sum^{\infty}_{n=0}a_{n}=s.$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E55 Encodings: TeX, pMML, png See also: Annotations for §1.15(viii), §1.15 and Ch.1

If

 1.15.56 $\lim_{x\to 1-}(1-x)\sum^{\infty}_{n=0}a_{n}x^{n}=s,$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E56 Encodings: TeX, pMML, png See also: Annotations for §1.15(viii), §1.15 and Ch.1

and either $|a_{n}|\leq K$ or $a_{n}\geq 0$, then

 1.15.57 $\lim_{n\to\infty}\frac{a_{0}+a_{1}+\dots+a_{n}}{n+1}=s.$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.15.E57 Encodings: TeX, pMML, png See also: Annotations for §1.15(viii), §1.15 and Ch.1