Prove the convergence of sequence √ 2, √ (2 + √ 2), √ (2 + √ (2 + √ 2)),. And find its limit

Prove the convergence of sequence √ 2, √ (2 + √ 2), √ (2 + √ (2 + √ 2)),. And find its limit

1. Increasing the proband sequence
It is obvious that the number sequence is increasing. We can also use the second mathematical induction to prove that the number sequence is increasing
Because A1 = √ 2

Prove the convergence of 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 that {x n} converges 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 & # 178; (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

Who can tell me about the expansion and reduction of sequence systematically,

The expansion and contraction method is actually a method in inequality. The so-called expansion and contraction method in sequence is a skill to prove sequence inequality
For example, to prove the sequence ∑ an ＜ a, we can first find ∑ BN ＜ a, find the BN satisfying the condition, and prove an ＜ BN, so as to prove that the process of ∑ an ＜ A is the expansion and contraction method of the sequence