دنبالههای حقیقی
Sequences
یکنوایی، کرانداری و همگرایی دنبالههای حقیقی.
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کارتها · 6
The order of a fixed-point iteration equals the index of the first non-vanishing derivative of g at the fixed point.
To solve f(x)=0 , iterate x_(n+1)=x_n-(f(x_n))/(f'(x_n)) — each step replaces f by its tangent line and takes its root; near a simple root it doubles the number of correct digits per step.
Pairing geometric and arithmetic means makes one sequence climb and the other fall; AM–GM traps them between each other, so both converge to a shared limit.
Any sequence built from a single geometric power rⁿ is decided by where r sits relative to ±1; for aₙ = rⁿ/(rⁿ+1) the limit is 0, ½, or 1 — after first checking the term is even defined.
A sequence given by x_(n+1)=f(x_n) converges once you show it is bounded and monotone; then the limit must be a fixed point f(L)=L.
For bounded sequences, limsup of a sum is at most the sum of limsups, and liminf is at least the sum of liminfs.