convergence

introduction

%%visits: 3 # intuition D’Alembert’s Criterion := $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n} |$ is less than 1 if ∑ⁿa_n is convergent and is greater than 1 if ∑ⁿa_n is divergent. It is inconclusive otherwise.

Convergence, every number in the sequence, eventually will stay within any given error window of the number they should become. # rigour A sequence a_nn = 1^∞ converges to a limit a ∈ ℝ, if ∀ϵ > 0∃N ∈ ℕ : ∀n ≥ N : |a_n − a| < ϵ monotone := non-increasing (all later terms are lower than or equal) or non-decreasing (all later terms are greater than or equal) Cauchy_convergence

Bolzano-Weierstrass theorem := Every bounded sequence has a convergent subsequence := Let x_nn = 1^∞ ⊂ [a, b]; then x_nn = 1^∞ has a limit point in [a, b] # examples tags :math:sequences_and_series:real_analysis:

backlinks

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