Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Lebl-Basic-Analysis-I · Problem 2.5.5
Sequences and Series
Mathematics · series-linearity
Problem For i = 1 , 2 , … , n i = 1, 2, \ldots, n i = 1 , 2 , … , n { x i , k } k = 1 ∞ \{x_{i,k}\}_{k=1}^{\infty} { x i , k } k = 1 ∞ n n n i ∈ N i \in \mathbb{N} i ∈ N ∑ k = 1 ∞ x i , k \sum_{k=1}^{\infty} x_{i,k} k = 1 ∑ ∞ x i , k ∑ i = 1 n ( ∑ k = 1 ∞ x i , k ) = ∑ k = 1 ∞ ( ∑ i = 1 n x i , k ) . \sum_{i=1}^{n} \left( \sum_{k=1}^{\infty} x_{i,k} \right) = \sum_{k=1}^{\infty} \left( \sum_{i=1}^{n} x_{i,k} \right). i = 1 ∑ n ( k = 1 ∑ ∞ x i , k ) = k = 1 ∑ ∞ ( i = 1 ∑ n x i , k ) . Browse the problem library