Some Theorems On the Summation of Divergent Series
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M" Combining (17) and (18) we have (19) l''(Ro+. . . +Rm-i)l/ni"l ^e, n>N', N' being equal to or greater than k either or N. Thus the theorem is proved. M— 1 From (12) and (14). (20) s— ;„S'„="[s+(s— s, ) + . . . + (s— s. . A)]l/m" ="(s+s+. .. +s)l/m»— »(0+s, + . .. +s, „. I)l/m'» =s— "(0+s, + . . . H-s^. I) l/m» where, „S„rr:(0+Si+. . . +Sm_i)l/m". Then (21) t«S„=:mS'„ or written out in full n(n±l)* (22) (n SiH s, + . . . +Sn(m. I) ) 1/m" 2! = [(m— l)a, + (m— 2)a, + . .. +a„_Jl/m+ n— 1 r (m— 1...) *(ai+. .. +a^) + *(a2+. .. +a„, ^i) + . .. +oo m, n— »oo That L, „S„=s follows from (21) and Theorem I. * n— >oo Consider L, „S„= L mS'n—s. The proof of Theorem I suffices for m— » 00 m— > 00 these cases provided we fix n and let m vary, replace the conditions n>N and n>N' by m>M and m>M' and note in connection with (18) that the first k The plus sign occurs for m>.
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To quickly assess the difficulty of the text, read a short excerpt:
M" Combining (17) and (18) we have (19) l''(Ro+. . . +Rm-i)l/ni"l ^e, n>N', N' being equal to or greater than k either or N. Thus the theorem is proved. M— 1 From (12) and (14). (20) s— ;„S'„="[s+(s— s, ) + . . . + (s— s. . A)]l/m" ="(s+s+. .. +s)l/m»— »(0+s, + . .. +s, „. I)l/m'» =s— "(0+s, + . . . H-s^. I) l/m» where, „S„rr:(0+Si+. . . +Sm_i)l/m". Then (21) t«S„=:mS'„ or written out in full n(n±l)* (22) (n SiH s, + . . . +Sn(m. I) ) 1/m" 2! = [(m— l)a, + (m— 2)a, + . .. +a„_Jl/m+ n— 1 r (m— 1...) *(ai+. .. +a^) + *(a2+. .. +a„, ^i) + . .. +oo m, n— »oo That L, „S„=s follows from (21) and Theorem I. * n— >oo Consider L, „S„= L mS'n—s. The proof of Theorem I suffices for m— » 00 m— > 00 these cases provided we fix n and let m vary, replace the conditions n>N and n>N' by m>M and m>M' and note in connection with (18) that the first k The plus sign occurs for m>.
What to read after Some Theorems On the Summation of Divergent Series?
You can find similar books in the "Read Also" column, or choose other free books by Glenn James to read onlineMoreLess
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