Sequences

=October 30th, 2012=



math a_1,a_2,a_3,...,a_n,... math

Can also be written as:

math ({a_n})_{n=1}^{\infty } math

**__Examples:__**
math 1,2,3,4,... \rightarrow a_n=n math

math 1,4,9,16,... \rightarrow a_n=n^2 math

math 1,2,4,8,16,32,... \rightarrow a_n=2^{n-1} math

math \frac{3}{5},\frac{-4}{25},\frac{5}{125},\frac{-6}{625} \rightarrow a_n=((-1)^{n+1})(\frac{n+2}{5^n}) math

__Example:__
math a_n=a_{n-1}+3 ;a_1=1 math

math 1,4,7,10,13,16,... math

math \rightarrow a_n=1+3(n-1) math

Recursively Defined Sequences
Definition: a sequence a_n has a limit L

math \lim_{n \to \infty } a_n=L math

OR

math a_n\rightarrow L math as math n\rightarrow \infty math

We can make the terms of a_n 'very' close to L by choosing n as 'very' large.

__Theorem:__
If

math \lim_{x\rightarrow \infty}f(x)=L math

and

math f(n)=a_n math

then...

math \lim_{n \to \infty } a_n=L math

__Example:__
math \lim_{n\rightarrow \infty} 1+(-1)^n math

Therefore,
 * n || 1 || 2 || 3 || 4 || odd || even ||
 * a_n || 0 || 2 || 0 || 2 || 0 || 2 ||

math \lim_{n\rightarrow \infty} 1+(-1)^n \rightarrow DNE math so it diverges.

More Examples:

 * 1. Give a general term for the following sequence:**


 * 7/2, 7/5, 7/8, 7/11, 1/2, 7/17,...**

math

First we rewrite 1/2 = 7/14, now we can recognize the denominator is increasing by 3 each time, while the numerator is always 7. The denominator sequence 2, 5, 8, 11, 12, 17 has a positive slope of 3 so the formula for the denominator is $3n+k$ for some $k$. To solve for k we test $n=1$ : $2 = 3*1+k$ so $k = -1$. Therefore the general term is $a_n=\frac{7}{3n-1}$.

math


 * 2. Give a recursive definition of the following sequence:**


 * 1, 4, 9, 16, 25, 36,...**

math

$

$a_n=1$ so we observe: $a_2=a_1+3, a_3=a_2+5, a_4=a_3+7, a_5=a_4+9$. The difference between consecutive terms are consecutive odd integers, so $s_n=s_(n-1)+2n-1$.

$

math

= Helpful Links: =

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 * 1. What is a Sequence? Basic Sequence Info (Video)**

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 * 2. Sequences - Examples showing convergence or divergence (Video)**

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 * 3. Some Special Limits to be Aware of**

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 * 4. Practice Problems - general sequences**

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 * 5. Practice Problems - arithmetic sequences**

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 * 6. Practice Problems - geometric sequences**