Problems and Theorems in Analysis pp 32-45 | Cite as

# Miscellaneous Problems

Chapter

## Abstract

We say that the series
are satisfied. The enveloping series may be convergent or divergent; if it converges its sum is the number

*a*_{0}+*a*_{1}+*a*_{2}+ ···*envelops*the number*A*if the relations$$|A - ({a_0} + {a_1} + ...{a_n})| < |{a_{n + 1}}|,n = 0,1,2...$$

*A.*Assume that*A, a*_{0},*a*_{1},*a*_{2}*...*are all real. If we have$$A - ({a_o} + {a_1} + {a_2} + ... + {a_n}) = {\theta _n}{a_{n + 1}},for{\kern 1pt} all{\kern 1pt} n = 0,1,2...and{\kern 1pt} 0 < {\theta _n} < 1$$

*A*is enveloped by the series*a*_{0}+*a*_{1}+*a*_{2}+ ···, and in fact it lies between two consecutive partial sums. In this situation we say the series is*enveloping A*in the*strict sense.*G. A. Scott and G. N. Watson [Quart. J. pure appl. Math. (London) Vol. 47, p. 312 (1917)] use the expression “arithmetically asymptotic’’ for a closely related concept. The terms of a strictly enveloping series have necessarily alternating signs.## Keywords

Strict Sense Stirling Number Maclaurin Series Arbitrary Complex Number Real Series
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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