Introduction to sequences

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  1. Overview:
  2. Notation of Sequences
  3. Definitions and theorems of Sequences
  1. Finding the terms of a sequence

    Find the first five terms of the following sequences.
    1. an=3(โˆ’1)n a_n=3(-1)^n
    2. ana_n= n+1n+1\frac{n+1}{\sqrt{n+1}}
    3. {cos(nฯ€2) cos(\frac{n\pi}{2}) }
  2. Finding the formula for a sequence

    Find the formula for the general term ana_n for the following sequences
    1. {12,13,14,15,... \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, ... }
    2. {12,25,38,411,... \frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, ... }
    3. {-1, 4, -9, 16, ... }
  3. Convergence and divergence of sequences

    Evaluate the limits and determine if the following limits are converging or diverging.
    1. limโก\limn →โˆž\infty (โˆ’1)nn2\frac{(-1)^n}{n^2}
    2. limโก\limn →โˆž\infty 6(12)n6(\frac{1}{2})^n
    3. limโก\limn →โˆž\infty n3+n+1n2+1\frac{n^3+n+1}{n^2+1}
Topic Notes
In this lesson, we will talk about what sequences are and how to formally write them. Then we will learn how to write the terms out of the sequences when given the general term. We will also learn how to write the general term when given a sequence. After learning the notations of sequences, we will take a look at the limits of sequences. Then we will take a look at some definitions and properties which will help us take the limits of complicating sequences. These theorems include the squeeze theorem, absolute value sequences, and geometric sequences.
1. If a sequence has the limit LL, then we can say that:

limโก\limn →โˆž\infty aann=L=L

If the limit is finite, then it is convergent. Otherwise, it is divergent.

2. If the limit of the sequences {ana_n} and {bnb_n} are finite and cc is constant, then we can say that

i) limโก\limn →โˆž\infty (an+bn)=limโก(a_n+b_n)=\limn →โˆž\infty an+a_n+limโก\limn →โˆž\infty bnb_n.
ii) limโก\limn →โˆž\infty (anโˆ’bn)=limโก(a_n-b_n)=\limn →โˆž\infty anโˆ’a_n-limโก\limn →โˆž\infty bnb_n.
iii) limโก\limn →โˆž\infty can=cca_n=c limโก\limn →โˆž\infty ana_n.
iv) limโก\limn →โˆž\infty(anbn)=(a_nb_n)= limโก\limn →โˆž\inftyanโˆ—a_n* limโก\limn →โˆž\infty bnb_n.
v) limโก\limn →โˆž\infty [an[a_nรท\divbn]b_n] =limโก=\limn →โˆž\inftyana_nรท\div limโก\limn →โˆž\inftybnb_n,, bnโ‰ 0b_n\neq0.

3. If anโ‰คcnโ‰คbna_n\leq c_n\leq b_n and limโก\limn →โˆž\infty an=a_n= limโก\limn →โˆž\infty bn=Lb_n=L, then limโก\limn →โˆž\infty cn=Lc_n=L.

4.if limโก\limn →โˆž\infty โˆฃanโˆฃ=0|a_n|=0, then limโก\limn →โˆž\infty an=0a_n=0 as well.

5. We say that:

Introduction to sequences

Where the sequence {xnx^n} is convergent for -1< xโ‰คx \leq 1, and divergent if xx > 1.