Miscellaneous Problems

  • George Pólya
  • Gabor Szegö
Part of the Springer Study Edition book series (SSE)

Abstract

We say that the series a 0 + a 1 + a 2 + ··· envelops the number A if the relations
$$|A - ({a_0} + {a_1} + ...{a_n})| < |{a_{n + 1}}|,n = 0,1,2...$$
are satisfied. The enveloping series may be convergent or divergent; if it converges its sum is A. Assume that A, a 0, a 1, a 2 ... are all real. If we have
$$A - ({a_o} + {a_1} + {a_2} + ... + {a_n}) = {\theta _n}{a_{n + 1}},for{\kern 1pt} all{\kern 1pt} n = 0,1,2...and{\kern 1pt} 0 < {\theta _n} < 1$$
the number 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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Copyright information

© Springer-Verlag Berlin Heidelberg 1972

Authors and Affiliations

  • George Pólya
    • 1
  • Gabor Szegö
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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