Collection
zero Useful+1
zero

Child column

Given sequence
Given the sequence {Xn}, arbitrarily select infinite items from it, and the sequence formed according to the original order is called a sub sequence of the sequence {Xn}.
The subsequence is a sequence of numbers, and it is convergent or divergent with any ordinary subsequence, and has the same limit when it converges.
Chinese name
Child column
Definition
A sequence of numbers formed by randomly selecting infinite terms from them in the original order

definition

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Definition 1: set up
Is a series , and
It is a strictly monotonic increase positive integer , then
,…,
,... also forms a sequence, called sequence
Child column of, recorded as
Definition 2: series
Itself and
The sub column obtained after removing the finite term is called
Ordinary sub columns of; A sub column that is not an ordinary sub column is called
The nontrivial child column of.

Subsequence theorem

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series
It is convergent or divergent with any ordinary subsequence, and has the same limit when it converges. [1]
series
Convergent Sufficient and necessary conditions Yes:
Any nontrivial subsequences of the converge to the same value. [2]