This paper describes a new method of obtaini ng an equivalent series from a given infi ni te co nv ergent or diverge nt series. I n many cases the new series is more convenient for su mming t.han the original and, moreover, the same method may usually be repeated indefinitely to obtain an entire sequence of series each equivalent t o t he original and each better than its predecessor in s umming propert ies. 'lh e new method differs from most s umming methods heretofore employed in that terms of t he transformed series are not linear function s of terms of the origin al series. The paper includes proofs of theor ems indicating the s cope of the new m ethod and comparisons of results with various other m ethods for many specific examples. GeneralInfinite series are not always amenable to convenient summing by usual methods. Even when convergent, a series may converge too slowly to permit ready evaluation or may consist of terms which, likely to offset any gain resulting from use of better fitting approximations. 3.I individually, require so much labor in evalu ation as to make determination of sufficient terms for summing even a moderately rapidly convergent series quite difficult. If th e series is asymptotic, the magnitude of the smallest term seems to set a lower bound on the accuracy of evaluation. For other types of divergent series, a convenient summing m ethod may not b e available. This paper presents a method of transforming series that may b e of value in all these cases. To t h e best of the author's knowledge, th e m ethod is origin al with him , and h as not been previously publish ed , although th e author has used it sin ce 1937. 1 Essentially, the method consists of approximating the series after the first m terms by a geometric series whose first term is the (m+ 1) th term of the original series and whose ratio is the ratio of the (m + 2)d term to the (m+ l )th. These approximating sums for successive values of m may b e considered to be su ccessive partial sums of a new series that has the same sum as the old but has, frequ ently, superior summing properties. Th e process may, in many cases, be repeated again on the new series to get a third series with still b etter properties. An indefinite number of repetitions may b e possible, as will be shown in examples given later.This method differs from summillg methods usua lly employed in that. th e terms of the transform ed series are not linear functions of the terms of the original series. It should :1.1so be pointed out that, various modifications of the method are possible such as use of other types of approximating series instead of the simple geometric series,2 but loss of s·mplici ty is 1 bince t ne nl'st draft of this paper, the aut hol' has learned of a talk "resented by D . Shanks at the Naval Ordnance Laboratory, White Oaks, Md . en t itled " M ath· em atical sequences treated as transients", w hich bas icall y considers each term of a series as th e sum of corresponding terms of one or more geometric series. ,~r h en a single geometric s...

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