Serie teleskopikoFrom Wikipedia, the free encyclopedia Matematikan, serie teleskopiko bat serie bat da non batura partzialek termino-kopuru finko bat duten ezeztatu ondoren. ( a 2 − a 1 ) + ( a 3 − a 2 ) + ( a 4 − a 3 ) + … + ( a n − a n − 1 ) = a n − a 1 {\displaystyle (a_{2}-a_{1})+(a_{3}-a_{2})+(a_{4}-a_{3})+\ldots +(a_{n}-a_{n-1})=a_{n}-a_{1}\,} Serie teleskopikoaren ohiko adibide bat Mengoliren seriea da, honela definitzen dena: ∑ n = 1 ∞ 1 n ( n + 1 ) {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}} honela kalkula daitekeena:[1] ∑ n = 1 ∞ 1 n ( n + 1 ) = ∑ n = 1 ∞ ( 1 n − 1 n + 1 ) = lim N → ∞ ∑ n = 1 N ( 1 n − 1 n + 1 ) = lim N → ∞ [ ( 1 − 1 2 ) + ( 1 2 − 1 3 ) + ⋯ + ( 1 N − 1 N + 1 ) ] = lim N → ∞ [ 1 + ( − 1 2 + 1 2 ) + ( − 1 3 + 1 3 ) + ⋯ + ( − 1 N + 1 N ) − 1 N + 1 ] = lim N → ∞ [ 1 − 1 N + 1 ] = 1. {\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}&{}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\sum _{n=1}^{N}\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\left\lbrack {\left(1-{\frac {1}{2}}\right)+\left({\frac {1}{2}}-{\frac {1}{3}}\right)+\cdots +\left({\frac {1}{N}}-{\frac {1}{N+1}}\right)}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1+\left(-{\frac {1}{2}}+{\frac {1}{2}}\right)+\left(-{\frac {1}{3}}+{\frac {1}{3}}\right)+\cdots +\left(-{\frac {1}{N}}+{\frac {1}{N}}\right)-{\frac {1}{N+1}}}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1-{\frac {1}{N+1}}}\right\rbrack =1.\end{aligned}}}
Matematikan, serie teleskopiko bat serie bat da non batura partzialek termino-kopuru finko bat duten ezeztatu ondoren. ( a 2 − a 1 ) + ( a 3 − a 2 ) + ( a 4 − a 3 ) + … + ( a n − a n − 1 ) = a n − a 1 {\displaystyle (a_{2}-a_{1})+(a_{3}-a_{2})+(a_{4}-a_{3})+\ldots +(a_{n}-a_{n-1})=a_{n}-a_{1}\,} Serie teleskopikoaren ohiko adibide bat Mengoliren seriea da, honela definitzen dena: ∑ n = 1 ∞ 1 n ( n + 1 ) {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}} honela kalkula daitekeena:[1] ∑ n = 1 ∞ 1 n ( n + 1 ) = ∑ n = 1 ∞ ( 1 n − 1 n + 1 ) = lim N → ∞ ∑ n = 1 N ( 1 n − 1 n + 1 ) = lim N → ∞ [ ( 1 − 1 2 ) + ( 1 2 − 1 3 ) + ⋯ + ( 1 N − 1 N + 1 ) ] = lim N → ∞ [ 1 + ( − 1 2 + 1 2 ) + ( − 1 3 + 1 3 ) + ⋯ + ( − 1 N + 1 N ) − 1 N + 1 ] = lim N → ∞ [ 1 − 1 N + 1 ] = 1. {\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}&{}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\sum _{n=1}^{N}\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\left\lbrack {\left(1-{\frac {1}{2}}\right)+\left({\frac {1}{2}}-{\frac {1}{3}}\right)+\cdots +\left({\frac {1}{N}}-{\frac {1}{N+1}}\right)}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1+\left(-{\frac {1}{2}}+{\frac {1}{2}}\right)+\left(-{\frac {1}{3}}+{\frac {1}{3}}\right)+\cdots +\left(-{\frac {1}{N}}+{\frac {1}{N}}\right)-{\frac {1}{N+1}}}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1-{\frac {1}{N+1}}}\right\rbrack =1.\end{aligned}}}