# Lecture 43: Sequences

• 5, 10, 15, 20, 25 ... {﻿${5n}$﻿} where ﻿$n= 1, 2, 3$﻿
• ﻿$f(1), f(2), f(3)$﻿ ...
• ﻿$a_1, a_2, a_3,$﻿ ... {﻿$a_n$﻿} 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 ﻿$n$﻿ increases the ﻿$n^{th}$﻿ term of a sequence gets close to a real number ﻿$L$﻿
• If a sequence {﻿$a_n$﻿} has a limit it is convergent, if not its divergent
1. ﻿$\lim_{x\rightarrow \infty} (a_n = b_n) = \lim_{x\rightarrow \infty}a_n \pm \lim_{x\rightarrow \infty} b_n$﻿
2. ﻿$\lim_{x\rightarrow \infty} ca_n = c\lim_{x\rightarrow \infty}a_n$﻿
3. ﻿$\lim_{x\rightarrow \infty} a_nb_n = \lim_{x\rightarrow \infty}a_n \lim_{x\rightarrow 0} b_n$﻿
4. ﻿$\lim_{x\rightarrow \infty} \frac {a_n}{b_n} = \frac {\lim_{x\rightarrow \infty}a_n }{\lim_{x\rightarrow \infty} b_n}$﻿ only if ﻿$\lim_{x\rightarrow \infty} b_n \neq 0$﻿
5. ﻿$\lim_{x\rightarrow \infty} c = c$﻿
6. ﻿$a_n < b_n < c_n \lim_{n \rightarrow \infty}a_n = \lim_{n \rightarrow \infty}c_n = L$﻿ therefore ﻿$\lim_{n \rightarrow \infty}b_n = L$﻿
• Sequence is increasing if ﻿$a_n < a_{n+1}$﻿
• If the sequence is bounded and monotonic (neither increasing or decreasing)

#### examples

1. ﻿$\lim_{n \rightarrow \infty} \frac{l_nn}{n} = 0$﻿
2. ﻿$\lim_{n \rightarrow \infty} \sqrt[n]{n} = 1$﻿
3. ﻿$\lim_{n \rightarrow \infty} x^{\frac{1}{n}} = 1$﻿
4. ﻿$\lim_{n \rightarrow \infty} x^n = 0$﻿
5. ﻿$\lim_{n \rightarrow \infty} (1+ \frac{x}{n})^n = e^x$﻿
6. ﻿$\lim_{n \rightarrow \infty} \frac{x^n}{n!} =0$﻿

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