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''for a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358501.png" />''
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''for a number $A$''
  
 
A series
 
A series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\sum_{n=0}^\infty a_n\label{*}\tag{*}$$
  
 
such that
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358503.png" /></td> </tr></table>
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$$|A-(a_0+\dotsb+a_n)|<|a_{n+1}|$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358504.png" />. An enveloping series may converge or diverge; if it converges, then its sum is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358505.png" />. The series (*) envelopes the real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358506.png" /> in the strict sense if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358507.png" /> are real and if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358508.png" />
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for all $n=0,1,\ldots$. An enveloping series may converge or diverge; if it converges, then its sum is equal to $A$. The series \eqref{*} envelopes the real number $A$ in the strict sense if the $a_n$ are real and if for all $n=0,1,\ldots,$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e0358509.png" /></td> </tr></table>
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$$A-(a_0+\dotsb+a_n)=\theta_na_{n+1},\quad0<\theta_n<1.$$
  
In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585010.png" /> lies between any two successive partial sums of the series. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585011.png" />, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585015.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585019.png" /> are enveloped in the strict sense by their MacLaurin series.
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In this case $A$ lies between any two successive partial sums of the series. For example, for $x>0$, the functions $e^{-x}$, $\ln(1+x)$, $(1+x)^{-p}$ ($p>0$), $\sin x$, $\cos x$, $\arctan x$, $J_0(x)$ are enveloped in the strict sense by their MacLaurin series.
  
If, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585020.png" />, the series
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If, for $x>R>0$, the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585021.png" /></td> </tr></table>
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$$\sum_{n=0}^\infty\frac{a_n}{x^n}$$
  
envelopes a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585022.png" /> taking real values, and if the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585023.png" /> are real, then the signs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585024.png" /> alternate and the series is enveloping in the strict sense. This series is an asymptotic expansion for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585025.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585026.png" />; if it is divergent, then it is called a semi-convergent series. Such series are used for the approximate computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585027.png" /> for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035850/e03585028.png" />.
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envelopes a function $f$ taking real values, and if the numbers $a_n$ are real, then the signs of $a_1,a_2,\ldots,$ alternate and the series is enveloping in the strict sense. This series is an asymptotic expansion for $f(x)$ as $x\to+\infty$; if it is divergent, then it is called a semi-convergent series. Such series are used for the approximate computation of $f(x)$ for large $x$.
  
 
====References====
 
====References====

Latest revision as of 15:50, 14 February 2020

for a number $A$

A series

$$\sum_{n=0}^\infty a_n\label{*}\tag{*}$$

such that

$$|A-(a_0+\dotsb+a_n)|<|a_{n+1}|$$

for all $n=0,1,\ldots$. An enveloping series may converge or diverge; if it converges, then its sum is equal to $A$. The series \eqref{*} envelopes the real number $A$ in the strict sense if the $a_n$ are real and if for all $n=0,1,\ldots,$

$$A-(a_0+\dotsb+a_n)=\theta_na_{n+1},\quad0<\theta_n<1.$$

In this case $A$ lies between any two successive partial sums of the series. For example, for $x>0$, the functions $e^{-x}$, $\ln(1+x)$, $(1+x)^{-p}$ ($p>0$), $\sin x$, $\cos x$, $\arctan x$, $J_0(x)$ are enveloped in the strict sense by their MacLaurin series.

If, for $x>R>0$, the series

$$\sum_{n=0}^\infty\frac{a_n}{x^n}$$

envelopes a function $f$ taking real values, and if the numbers $a_n$ are real, then the signs of $a_1,a_2,\ldots,$ alternate and the series is enveloping in the strict sense. This series is an asymptotic expansion for $f(x)$ as $x\to+\infty$; if it is divergent, then it is called a semi-convergent series. Such series are used for the approximate computation of $f(x)$ for large $x$.

References

[1] G. Pólya, G. Szegö, "Problems and theorems in analysis" , Springer (1976) pp. Chapts. 1–2 (Translated from German)
[2] G.H. Hardy, "Divergent series" , Clarendon Press (1949)


Comments

References

[a1] G.A. Scott, G.N. Watson, "Asymptotic formula occurring in electron theory" Quart. J. Math. , 47 (1917) pp. 312
How to Cite This Entry:
Enveloping series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Enveloping_series&oldid=15647
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article