Extended data
Divergence function
In mathematical analysis, the concept opposite to convergence is divergence. Divergent Series refers to non-convergent series (in Cauchy sense). Such as series
and
In other words, there is no finite limit for any part or sequence of this sequence.
If a series is convergent, the terms of this series will definitely tend to zero. Therefore, any series whose term does not tend to zero is divergent. However, convergence is a stronger requirement than this: not every series whose term tends to zero converges. One of the counterexamples is harmonic series.
Medieval mathematician oris Mie proved the divergence of harmonic series.
Convergence function:
Given a function sequence defined on the interval I, u 1(x), u2(x), u3(x) ... to the United Nations (x) ... and then the expression U 1 (x)+U2 (x)+U3 (x)+
For each determined value X0∈I, the function term series (1) becomes a constant term series U1(x0)+U2 (x0)+U3 (x0)+...+UN (x0)+... (2) This series can converge or diverge. If the series (2) diverges, the point x0 is called the divergence point of the series of function terms (1).
The whole convergence point of the function term series (1) is called its convergence domain, and the whole divergence point is called its divergence domain, corresponding to any number x in the convergence domain.
Function term series is called convergent constant term series, so there is a definite sum S. In this way, in the convergent domain, the sum of function term series is the function S(x) of x, usually called the sum function of function term series, and the definition domain of this function is the convergent domain of series.
And it is written as s (x) = u 1 (x)+U2 (x)+u3 (x)+UN (x)+... If the sum of the first n terms of the function term series (1) is Sn(x), there is lim n →∞ sn (x) in the convergence domain.
Remember that rn(x) = s (x)-sn (x), and rn(x) is called the remainder of the function series term (of course, only x is meaningful in the convergence domain, lim n→∞rn (x)=0.
References:
Baidu encyclopedia-divergence function