Enveloping series

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)

References

[a1]	G.A. Scott, G.N. Watson, "Asymptotic formula occurring in electron theory" Quart. J. Math. , 47 (1917) pp. 312