A basic problem to prove the limit of sequence: LIM (n - > infinite) 3N / (n + 1) = 3... How to prove it? The answer is n = [(3 / x) - 1] x is any small positive number

A basic problem to prove the limit of sequence: LIM (n - > infinite) 3N / (n + 1) = 3... How to prove it? The answer is n = [(3 / x) - 1] x is any small positive number

be careful:
The proof and calculation of limit are different. If the algorithm of limit is used, it is the same as upstairs, and the numerator and denominator are changed into the form of solvable limit
However, if it is proof, the strict e-N definition shall be used, but it is not required in high school

According to the definition of sequence limit, it is proved that LIM (x →∞) (3N + 1) / (2n-1) = 3 / 2

The definition method of the standard proves that:
Hope to adopt!

Find the answer sequence {x} of a high number problem is bounded, LIM (n →∞) y = 0, and prove LIM (n →∞) xy = 0

Because xn is bounded, │ xn │≤ M. where m is a positive number. And because limyn = 0 (n tends to infinity), for any positive number ε, There is a positive number n, when x > N, │ yn │

Prove the convergence of the sequence and find the limit Let a > 0, 0 < X1 < 1 / A, x n + 1 = x n (2 - A * x n) (n = 1,2,...). Prove the convergence of {x n}, and find LIM (n → 0) xn

Xn+1=Xn × (2-a*Xn)=-a × (Xn-1/a) ²+ 1/a
→ (1/a-Xn+1)=a × (1/a-Xn) ²
Let yn = 1 / a-xn, then yn + 1 = a × Yn ² (Y1=1/a-X1,n≥2)
∴Yn+1=a^(2*n-1) × Y1^(2*n)=1/a × (a*Y1)^(2*n)
∴Xn+1=1/a-1/a × (a*Y1)^(2*n)
∵ Y1 = 1 / a-x1, that is, 0 ＜ Y1 ＜ 1 / A
∴0＜a*Y1＜1
∴0＜(a*Y1)^(2*n)＜1
∴0＜Xn+1＜1/a
When n → + ∞, (a * Y1) ^ (2 * n) → 0, xn + 1 → 1 / A

It is proved that if the sequence converges to a, any of its subsequences also converges, and the limit is also a Can that be expressed in mathematical language?

Counter evidence can be used
If the subsequence does not converge or the convergence limit is a, the original sequence does not converge

Let X1 = a > 0, xn + 1 = 1 / 2 (xn + 2 / xn), n = 1,2,3... Prove the convergence of sequence {xn} by using monotone bounded criterion Such as title

Bounded: xn + 1 = 1 / 2 (xn + 2 / xn) > = 1 / 2 * 2 * root sign (xn * 2 / xn) = root sign 2 n = 1,2,3
Monotone: xn + 1-xn = - 1 / 2 (xn-2 / xn) when n > = 2, xn > = root 2, so xn + 1-xn