What is the relationship between any value of limit and n in higher mathematics?

First, choose an arbitrarily small positive number ε. For this ε, which is selected as a constant value, if its nth term can be found in the sequence {xn}, the terms after the nth term in the sequence (i.e. n >；; N) all satisfy the inequality |xn-a|N (for example, n= 100 1, 1002 ...).

For any given ε, there is n, which is actually a function of ε, so some books write it as n (ε). Note that with the change of ε, n can naturally change accordingly.

Expressed in logical language, that is, for any small ε >; 0 (used to describe the degree of proximity), with a certain n, when n >; When n is backward enough, the absolute value of the difference between the sequence value and the limit value is less than ε (as small as we expected in advance).

Modern mathematics pays great attention to language, so you need to have a deep understanding of "mathematical language". In order to do this, we should do exercises properly and have a certain degree of psychological maturity.

Extended data

For example:

It is known that for any positive integer n, there is a1+A2+...+An = N 3, then lim [1(A2-1)+1(A3-1)+...+60.

Solution: n>2 o'clock, a( 1)+a(2)+a(3)+. . . . . . +a(n- 1)

=(n- 1)^3。 Subtract the original formula with this formula to get a (n) = n 3-(n-1) 3 = 3n 2-3n+1.

Therefore,1(a (n)-1) =1(3N2-3N) =1/3 (n-1)-1/3n.

Thus1/(a (2)-1)+1/(a (3)-1)+1(a (4)-1)+. . . . . . + 1/(a(n)- 1)

=( 1/3- 1/6)+( 1/6- 1/9)+( 1/9- 1/ 12)+。 . . . . . +( 1/3(n- 1)- 1/(3n))

= 1/3- 1/3n from which the original formula can be obtained.

= lim(n→+∞)( 1/3- 1/3n)

= 1/32。 Solution:

lim(n→+∞)na(n)