By G. Fichtenholz
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Extra resources for Infinite Series: Ramifications (Pocket Mathematical Library)
THEOREM 2. Suppose the iterated series (3) is absolutely convergent. Then (3) is convergent, and the series (6) made up of the elements of (1) arranged in any order whatsoever is absolutely convergent, with the same suns as (3). 3 Fichtenholz(2094) Infinite Series: Ramifications 34 By hypothesis, the series (9) is convergent. Let A* be its sum. Then, given any m and n, we have [n ``m L L. +lurl of the series (6*). , u, all belong to the first m columns and the first n rows of the matrix (1) if m and n are sufficiently large.
Then, given any m and n, we have [n ``m L L. +lurl of the series (6*). , u, all belong to the first m columns and the first n rows of the matrix (1) if m and n are sufficiently large. , (6) is absolutely convergent. The rest of the proof is now an immediate consequence of Theorem 1. Remark. Clearly, Theorems 1 and 2 remain true if (3) is replaced by (3'). THEOREM 3. Given a matrix (1), suppose the iterated series (3) is absolutely convergent. Then the iterated series (3') converges and has the same sum as (3) : m n j=1k=1 n ajk' m k=1j=1 ask ) Proof.
I J(;W(M- I 1 + m2 (m2 ` 1) + m3 (m3 - 1) + 1) (justify the various rearrangements). It follows that - n=2 n (n G_ °° 1 where n now ranges over all positive integers starting from 2. Therefore OD G2: ( n=2 n - I 1 1 n Example 3. Consider the matrix with general term (j -_ 1)! (k - 1)! (k+j) (where 0! = 1). Setting a = 0, p = k in the formula 00 1 Y. , Prob. 2, p. 4), we can easily sum the terms of the kth row : OD a(k) = (k k - k! k2 Hence the sum of one iterated series is 00 k 00 CO J=1 , 1 () k k Because of the symmetry of a() with respect to j and k, the other iterated series is identical with the first, and nothing new can be deduced by equating the sums of the two iterated series.
Infinite Series: Ramifications (Pocket Mathematical Library) by G. Fichtenholz