EngliSea > M > math > 55 Analysis > 3 Sequences
 
Proverbs and Quotes in English
Short sentences and easy words.
 
30 approximating - √2, φ
9707#20533 SIBLINGS CHILDREN 20533
 
311 Limit (Sequence)
A sequence 「aₙ」 converges to a limit L if and only if for every ε>0 there exists a natural number N such that n>N ⇒ ∣aₙ−L∣<ε.
9707#9704 SIBLINGS CHILDREN COMMENT 9704
 
312 Rules (Sequence)
Suppose that 「aₙ」 and 「bₙ」 are convergent sequences with limits A and B respectively; then the following rules apply: ❶ Sum rule 「aₙ+bₙ」 converges to A+B. ❷ Product rule 「aₙ·bₙ」 converges to A·B. ❸ Quotient rule 「aₙ/bₙ」 converges to A/B, provided that ...
9707#3775 SIBLINGS CHILDREN COMMENT 3775
 
313 Sandwich rule (Sequence)
Let 「aₙ」, 「bₙ」 and 「cₙ」 be sequences satisfying aₙ≤bₙ≤cₙ for all n∈ℕ. If 「aₙ」 and 「cₙ」 both converges to the same limit L then 「bₙ」 also converges to L.
9707#9718 SIBLINGS CHILDREN COMMENT 9718
 
314 Composite rule (Sequence)
Let 「aₙ」 be a convergent sequence with limit L and let f be a continuous function whose domain contains 「aₙ」 and L. Then the sequence 「f(aₙ)」 converges to f(L).
9707#9719 SIBLINGS CHILDREN COMMENT 9719
 
32 Expansion of Operators and Numbers
32 Expansion of Operators and Numbers
「n˄p」 and 「c˄n」 are on the graph of y=x˄p and y=c˄x respectively.
9707#9723 SIBLINGS CHILDREN COMMENT 9723
 
321 Power rule (Sequence)
If 「aₙ」 is a null sequence where aₙ≥0 for all n∈ℕ, and if p∈ℝ and p>0, then 「aₙ˄p」 is a null sequence.
9707#9720 SIBLINGS CHILDREN COMMENT 9720
 
322 Basic null sequences
ᐥThe following are null sequences. ❶ 「1/n˄p」 for p>0 ❷ 「c˄n」 for |c|<1 ❸ 「n˄p·c˄n」 for p>0 and |c|<1 ❹ 「c˄n/n!」 for c∈ℝ ❺ 「n˄p/n!」 for p>0ᐥ Proof of ❶ ∀ε>0 ∃N∈ℕ such that N > 1/ε. n > N ⇒ n > 1/ε ⇒ 1/n < ε ⇒ |1/n−0| < ε 「1/n」 is a null sequence. By the p ...
9707#9722 SIBLINGS CHILDREN 9722
 
322_ Binomial Expansion
ᐥ(a+b)˄n = 「「nꞒk」·a˄(n−k)·b˄k Σk=0,n」 = 「n!/((n−k)!·k!)·a˄(n−k)·b˄k Σk=0,n」ᐥ (a+b)˄1 = 「1!/((1−k)!·k!)·a˄(1−k)·b˄k Σk=0,1」 = a+b (a+b)˄2 = 「2!/((2−k)!·k!)·a˄(2−k)·b˄k Σk=0,2」 = a˄2+2·a·b+b˄2 (a+b)˄(n+1) = (a+b)˄n·(a+b) = (a+b)˄n·a+(a+b)˄n·b = 「n!/((n−k ...
9707#9688 SIBLINGS CHILDREN COMMENT 9688
 
335 Subsequence
ᐥThe sequence 「a⸤n⸤r⸥⸥」 is a subsequence of the sequence 「a⸤n⸥」 if 「n⸤r⸥」 is a strictly increasing sequence of natural numbers. 「a⸤n⸥ Ƚn→∞」 = L ⇒ 「a⸤n⸤r⸥⸥ Ƚn→∞」 = L 「a⸤n⸥ Ƚn→∞」 = ∞ ⇒ 「a⸤n⸤r⸥⸥ Ƚn→∞」 = ∞ᐥ
9707#5220 SIBLINGS CHILDREN 5220
 
341 Principle of monotone sequences
A bounded monotone sequence is convergent.
9707#9766 SIBLINGS CHILDREN 9766
 
342 Bonzano-Weierstrass theorem
A subset of Rⁿ is sequentially compact if and only if it is closed and bounded.
9707#5120 SIBLINGS CHILDREN 5120
 
342 Bonzano-Weierstrass theorem
Any bounded sequence of real numbers contains a convergent subsequence.
9707#9765 SIBLINGS CHILDREN 9765
 
343 Cauchy sequence
ᐥA sequence 「aₙ」 is a Cauchy sequence if and only if for every ε>0 there exists a natural number N such that n,m>N ⇒ |a⸤n⸥−a⸤m⸥|<ε. A sequence is convergent if and only if the sequence is a Cauchy sequence.ᐥ If 「a⸤n⸥」 is a convergent sequence, then for a ...
9707#5221 SIBLINGS CHILDREN 5221
 
34_ 「(1+1/n)˄n」 is convergent
(1+1/n)˄n = 「n!/((n−i)!·i!)·1˄(n−i)·(1/n)˄i Σi=0,n」 = 「(1/i!)·n!/(n−i)!·(1/n˄i) Σi=0,n」 = 「(1/i!)·「(n−j) Πj=0,i−1」·(1/n˄i) Σi=0,n」 = 「(1/i!)·「(n−j)/n Πj=0,i−1」 Σi=0,n」 = 「(1/i!)·「(1−j/n) Πj=0,i−1」 Σi=0,n」 < 「(1/i!)·「(1−j/(n+1)) Πj=0,i−1」 Σi=0,n」 < 「(1/i!) ...
9707#9767 SIBLINGS CHILDREN 9767
 
harmonic numbers
9707#8621 SIBLINGS CHILDREN 8621

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