數列的夾擠定理示例
Author: 藍邦偉  老師

數列的夾擠定理示例

$ 已知數列c_n滿足(n+1)²-n²≥nc_n≥n²-(n-1)²\;，\;試求數列\left\langle c_n  \right\rangle 的極限值 $

$ Sol： $

$ ∵2n+1≥nc_n≥2n-1 $

$ ∴\frac{2n+1}{n}≥c_n≥\frac{2n-1}{n} $

$ 又\lim_{ n \to \infty } \frac{2n+1}{n}=\lim_{ n \to \infty }\frac{2n-1}{n}=2$

$ ∴\lim_{ n \to \infty }c_n=2 $

$ 設a_n=(n+1)²-n²=2n+1，\;b_n=n²-(n-1)²=2n-1 $

$a_n  $

$ b_n $

$ c_n $

$ y=2 $

n = 1 (slider from 1 to 1000, step 1)
List_1 = Sequence((2 * k + 1) / k, k, 1, n)
List_2 = Sequence((2 * k - 1) / k, k, 1, n)
Step = 0 (slider from 0 to 7, step 1)
A = (-1, 0)
B: A + (7, 0)
u = Vector(A, B)
O = Point(u)
c = Circle(O, A)
d = Circle(O, B)
a: x = x(O)
C = Intersect(d, a, 1)
D = Intersect(c, a, 2)
v = Vector(D, C)
b = Segment((0, 2), (6, 2))
List_a = Sequence((k, 2 + 1 / k), k, 1, 6)
List_b = Sequence((k, 2 - 1 / k), k, 1, 6)
text6: "$說明：\; a_{" + (LaTeX[n]) + "}=" + (LaTeX[FractionText[Element[List_1, n]]]) + "=" + (LaTeX[Element[List_1, n]]) + " ≥c_{" + (LaTeX[n]) + "}≥" + (LaTeX[Element[List_2, n]]) + "=" + (LaTeX[FractionText[Element[List_2, n]]]) + "=b_{" + (LaTeX[n]) + "}$"
List_c = Sequence((k, 0.1 * sin(k) + 2), k, 1, 6)