# §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, $\mathrm{i}$: imaginary unit 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, $\mathrm{i}$: imaginary unit, $n$: 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, $\mathrm{i}$: imaginary unit, $\int$: 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 $s_{n}(\theta)=\sum^{n}_{k=-n}F(k)e^{ik\theta}.$ ⓘ

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, $\mathrm{i}$: imaginary unit, $n$: 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, $\mathrm{i}$: imaginary unit, $n$: 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 $G(z)=G(x+iy)=u(x,y)+iv(x,y)=F(0)+2\sum^{\infty}_{n=1}F(n)z^{n}.$

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 (V1.0.14) 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 (V1.0.14) 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