Lecture 43: Sequences

  • 5, 10, 15, 20, 25 ... {5n{5n}} where n=1,2,3n= 1, 2, 3
  • f(1),f(2),f(3)f(1), f(2), f(3)  ...
  • a1,a2,a3,a_1, a_2, a_3,  ... {ana_n} in general
  • An infinite sequence is a function where domain is the set of positive integers and range is a set of real numbers
  • As nn increases the nthn^{th} term of a sequence gets close to a real number LL 
  • If a sequence {ana_n} has a limit it is convergent, if not its divergent
  1. limx(an=bn)=limxan±limxbn\lim_{x\rightarrow \infty} (a_n = b_n) = \lim_{x\rightarrow \infty}a_n \pm \lim_{x\rightarrow \infty} b_n
  2. limxcan=climxan\lim_{x\rightarrow \infty} ca_n = c\lim_{x\rightarrow \infty}a_n 
  3. limxanbn=limxanlimx0bn\lim_{x\rightarrow \infty} a_nb_n = \lim_{x\rightarrow \infty}a_n \lim_{x\rightarrow 0} b_n
  4. limxanbn=limxanlimxbn\lim_{x\rightarrow \infty} \frac {a_n}{b_n} = \frac {\lim_{x\rightarrow \infty}a_n }{\lim_{x\rightarrow \infty} b_n} only if limxbn0\lim_{x\rightarrow \infty} b_n \neq 0
  5. limxc=c\lim_{x\rightarrow \infty} c = c
  6. an<bn<cnlimnan=limncn=La_n < b_n < c_n \lim_{n \rightarrow \infty}a_n = \lim_{n \rightarrow \infty}c_n = L therefore limnbn=L \lim_{n \rightarrow \infty}b_n = L
  • Sequence is increasing if an<an+1a_n < a_{n+1}
  • If the sequence is bounded and monotonic (neither increasing or decreasing)


examples

  1. limnlnnn=0\lim_{n \rightarrow \infty} \frac{l_nn}{n} = 0
  2. limnnn=1\lim_{n \rightarrow \infty} \sqrt[n]{n} = 1 
  3. limnx1n=1\lim_{n \rightarrow \infty} x^{\frac{1}{n}} = 1
  4. limnxn=0\lim_{n \rightarrow \infty} x^n = 0
  5. limn(1+xn)n=ex\lim_{n \rightarrow \infty} (1+ \frac{x}{n})^n = e^x
  6. limnxnn!=0\lim_{n \rightarrow \infty} \frac{x^n}{n!} =0 


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