求无穷级数之和 ∑(n=1,∞)1／(1^2+2^2+…+n^2)

elim

王守恩老师的有关a的结果很可能是对的．论证留给大家．

王守恩

永远 发表于 2021-1-24 16:59
楼上小学的裂项法……～～

小学的裂项法……，怎么裂？

\(\frac{1}{2*3}+\frac{1}{5*6}+\frac{1}{9*10}+\frac{1}{14*15}+\frac{1}{20*21}+\frac{1}{27*28}+\frac{1}{35*36}+\frac{1}{44*45}+\frac{1}{54*55}+\cdots=\frac{2}{9}\)

王守恩

永远 发表于 2021-1-24 16:59
楼上小学的裂项法……～～

震撼的数阵。
\(\frac{1}{2*3}+\frac{1}{5*6}+\frac{1}{9*10}+\frac{1}{14*15}+\frac{1}{20*21}+\frac{1}{27*28}+\frac{1}{35*36}+\frac{1}{44*45}+\frac{1}{54*55}+\cdots=\frac{2}{9}\)
\(\frac{1}{2*3}+\frac{2}{5*6}+\frac{3}{9*10}+\frac{4}{14*15}+\frac{5}{20*21}+\frac{6}{27*28}+\frac{7}{35*36}+\frac{8}{44*45}+\frac{9}{54*55}+\cdots=\frac{3}{9}\)
\(\frac{2}{2*3}+\frac{3}{5*6}+\frac{4}{9*10}+\frac{5}{14*15}+\frac{6}{20*21}+\frac{7}{27*28}+\frac{8}{35*36}+\frac{9}{44*45}+\frac{10}{54*55}+\cdots=\frac{5}{9}\)
\(\frac{3}{2*3}+\frac{4}{5*6}+\frac{5}{9*10}+\frac{6}{14*15}+\frac{7}{20*21}+\frac{8}{27*28}+\frac{9}{35*36}+\frac{10}{44*45}+\frac{11}{54*55}+\cdots=\frac{7}{9}\)
\(\frac{4}{2*3}+\frac{5}{5*6}+\frac{6}{9*10}+\frac{7}{14*15}+\frac{8}{20*21}+\frac{9}{27*28}+\frac{10}{35*36}+\frac{11}{44*45}+\frac{12}{54*55}+\cdots=\frac{9}{9}\)

王守恩

永远 发表于 2021-1-24 16:59
楼上小学的裂项法……～～

这不是挺好玩吗？！
\(\frac{0}{5*6}+\frac{1}{9*10}+\frac{2}{14*15}+\frac{3}{20*21}+\frac{4}{27*28}+\frac{5}{35*36}+\frac{6}{44*45}+\frac{7}{54*55}+\cdots=\frac{1}{18}\)
\(\frac{1}{5*6}+\frac{2}{9*10}+\frac{3}{14*15}+\frac{4}{20*21}+\frac{5}{27*28}+\frac{6}{35*36}+\frac{7}{44*45}+\frac{8}{54*55}+\cdots=\frac{2}{18}\)
\(\frac{2}{5*6}+\frac{3}{9*10}+\frac{4}{14*15}+\frac{5}{20*21}+\frac{6}{27*28}+\frac{7}{35*36}+\frac{8}{44*45}+\frac{9}{54*55}+\cdots=\frac{3}{18}\)
\(\frac{3}{5*6}+\frac{4}{9*10}+\frac{5}{14*15}+\frac{6}{20*21}+\frac{7}{27*28}+\frac{8}{35*36}+\frac{9}{44*45}+\frac{10}{54*55}+\cdots=\frac{4}{18}\)
\(\frac{4}{5*6}+\frac{5}{9*10}+\frac{6}{14*15}+\frac{7}{20*21}+\frac{8}{27*28}+\frac{9}{35*36}+\frac{10}{44*45}+\frac{11}{54*55}+\cdots=\frac{5}{18}\)

王守恩

永远 发表于 2021-1-24 16:59
楼上小学的裂项法……～～

数学有那么神秘吗？！
\(\frac{1}{1*6}+\frac{1}{3*10}+\frac{1}{6*15}+\frac{1}{10*21}+\frac{1}{15*28}+\frac{1}{21*36}+\frac{1}{28*45}+\frac{1}{36*55}+\cdots=\frac{2}{9}\)
\(\frac{1}{1*6}+\frac{2}{3*10}+\frac{3}{6*15}+\frac{4}{10*21}+\frac{5}{15*28}+\frac{6}{21*36}+\frac{7}{28*45}+\frac{8}{36*55}+\cdots=\frac{3}{9}\)
\(\frac{2}{1*6}+\frac{2}{3*10}+\frac{2}{6*15}+\frac{2}{10*21}+\frac{2}{15*28}+\frac{2}{21*36}+\frac{2}{28*45}+\frac{2}{36*55}+\cdots=\frac{4}{9}\)
\(\frac{2}{1*6}+\frac{3}{3*10}+\frac{4}{6*15}+\frac{5}{10*21}+\frac{6}{15*28}+\frac{7}{21*36}+\frac{8}{28*45}+\frac{9}{36*55}+\cdots=\frac{5}{9}\)
\(\frac{3}{1*6}+\frac{3}{3*10}+\frac{3}{6*15}+\frac{3}{10*21}+\frac{3}{15*28}+\frac{3}{21*36}+\frac{3}{28*45}+\frac{3}{36*55}+\cdots=\frac{6}{9}\)
\(\frac{3}{1*6}+\frac{4}{3*10}+\frac{5}{6*15}+\frac{6}{10*21}+\frac{7}{15*28}+\frac{8}{21*36}+\frac{9}{28*45}+\frac{10}{36*55}+\cdots=\frac{7}{9}\)

王守恩

永远 发表于 2021-1-24 16:59
楼上小学的裂项法……～～

动手就简单！
\(\frac{3}{9*10}+\frac{4}{14*15}+\frac{5}{20*21}+\frac{6}{27*28}+\frac{7}{35*36}+\frac{8}{44*45}+\frac{9}{54*55}+\frac{10}{65*66}+\frac{11}{77*78}+\cdots=\frac{1}{10}\)
\(\frac{9}{9*10}+\frac{9}{14*15}+\frac{9}{20*21}+\frac{9}{27*28}+\frac{9}{35*36}+\frac{9}{44*45}+\frac{9}{54*55}+\frac{9}{65*66}+\frac{9}{77*78}+\cdots=\frac{2}{10}\)
\(\frac{12}{9*10}+\frac{13}{14*15}+\frac{14}{20*21}+\frac{15}{27*28}+\frac{16}{35*36}+\frac{17}{44*45}+\frac{18}{54*55}+\frac{19}{65*66}+\frac{20}{77*78}+\cdots=\frac{3}{10}\)
\(\frac{18}{9*10}+\frac{18}{14*15}+\frac{18}{20*21}+\frac{18}{27*28}+\frac{18}{35*36}+\frac{18}{44*45}+\frac{18}{54*55}+\frac{18}{65*66}+\frac{18}{77*78}+\cdots=\frac{4}{10}\)
\(\frac{21}{9*10}+\frac{22}{14*15}+\frac{23}{20*21}+\frac{24}{27*28}+\frac{25}{35*36}+\frac{26}{44*45}+\frac{27}{54*55}+\frac{28}{65*66}+\frac{29}{77*78}+\cdots=\frac{5}{10}\)
\(\frac{27}{9*10}+\frac{27}{14*15}+\frac{27}{20*21}+\frac{27}{27*28}+\frac{27}{35*36}+\frac{27}{44*45}+\frac{27}{54*55}+\frac{27}{65*66}+\frac{27}{77*78}+\cdots=\frac{6}{10}\)
\(\frac{30}{9*10}+\frac{31}{14*15}+\frac{32}{20*21}+\frac{33}{27*28}+\frac{34}{35*36}+\frac{35}{44*45}+\frac{36}{54*55}+\frac{37}{65*66}+\frac{38}{77*78}+\cdots=\frac{7}{10}\)
\(\frac{36}{9*10}+\frac{36}{14*15}+\frac{36}{20*21}+\frac{36}{27*28}+\frac{36}{35*36}+\frac{36}{44*45}+\frac{36}{54*55}+\frac{36}{65*66}+\frac{36}{77*78}+\cdots=\frac{8}{10}\)
\(\frac{39}{9*10}+\frac{40}{14*15}+\frac{41}{20*21}+\frac{42}{27*28}+\frac{43}{35*36}+\frac{44}{44*45}+\frac{45}{54*55}+\frac{46}{65*66}+\frac{47}{77*78}+\cdots=\frac{9}{10}\)