Enveloping series

for a number

A series

(*)

such that

for all . An enveloping series may converge or diverge; if it converges, then its sum is equal to . The series (*) envelopes the real number in the strict sense if the are real and if for all

In this case lies between any two successive partial sums of the series. For example, for , the functions , , (), , , , are enveloped in the strict sense by their MacLaurin series.

If, for , the series

envelopes a function taking real values, and if the numbers are real, then the signs of alternate and the series is enveloping in the strict sense. This series is an asymptotic expansion for as ; if it is divergent, then it is called a semi-convergent series. Such series are used for the approximate computation of for large .

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