# 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.

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