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Get Infinite Series: Ramifications (Pocket Mathematical Library) PDF

Posted On April 20, 2018 at 12:24 pm by / Comments Off on Get Infinite Series: Ramifications (Pocket Mathematical Library) PDF

By G. Fichtenholz

ISBN-10: 0677209401

ISBN-13: 9780677209401

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Example text

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. Suppose we now modify the matrix as follows: We retain the first k -- 1 terms in the kth row, but replace the kth term by the sum of all the terms of the kth row starting from the kth, dropping the remaining terms altogether.

5. 8. Double Series There is another kind of series associated with the infinite rectangular matrix (1), p. 30, namely the series of the form ail) + aZl) + ... l) + ... +;Z)+... +a2)+... +a" + ... k= 1 (with only one summation sign). By a partial sum of (1), we mean a finite sum of the form 111 n An. J=1k=1 aJ made up of the terms in the first m columns and the first n rows of the given matrix. Let m and n approach infinity independently. Ft N -M 2. The limit in (2) is a double limit. Thus if A is finite, (2) means that, given any e > 0, there is an integer N such that Ann) Whenever m and n both exceed N.

Yn Y1 n Then the first term on the right approaches zero, for the reason just given, while the second approaches ab, by Corollary 2. 0 PROBLEMS 1. Given two sequences {x,,} and {y,,}, suppose that a) {y,,} is an increasing sequence with limit + cc ; b) lim X,. - Xn- n-- Y. 1 Yn-1 =a (x0 = Yo = 0) . Operations on Series 25 Prove that - = a. lira X. n- m Y. Hint. Choose tnm = Ym-Ym-1 Yn 2. , let xn = Coxo + C;x1 + C2"x2 + + C,",xn f 2" where Cm is the binomial coefficient n! , n). m! (n-m)!

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Infinite Series: Ramifications (Pocket Mathematical Library) by G. Fichtenholz

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