Introduction to sequences

Introduction to sequences

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:

lim\limn →\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) lim\limn →\infty (an+bn)=lim(a_n+b_n)=\limn →\infty an+a_n+lim\limn →\infty bnb_n.
ii) lim\limn →\infty (anbn)=lim(a_n-b_n)=\limn →\infty ana_n-lim\limn →\infty bnb_n.
iii) lim\limn →\infty can=cca_n=c lim\limn →\infty ana_n.
iv) lim\limn →\infty(anbn)=(a_nb_n)= lim\limn →\inftyana_n* lim\limn →\infty bnb_n.
v) lim\limn →\infty [an[a_n÷\divbn]b_n] =lim=\limn →\inftyana_n÷\div lim\limn →\inftybnb_n,, bn0b_n\neq0.

3. If ancnbna_n\leq c_n\leq b_n and lim\limn →\infty an=a_n= lim\limn →\infty bn=Lb_n=L, then lim\limn →\infty cn=Lc_n=L.

4.if lim\limn →\infty an=0|a_n|=0, then lim\limn →\infty an=0a_n=0 as well.

5. We say that:

Introduction to sequences

Where the sequence {xnx^n} is convergent for -1< xx \leq 1, and divergent if xx > 1.
  • Introduction
    Notation of Sequences

    Definitions and theorems of Sequences

  • 1.
    Finding the terms of a sequence

    Find the first five terms of the following sequences.
    an=3(1)n a_n=3(-1)^n

    ana_n= n+1n+1\frac{n+1}{\sqrt{n+1}}

    {cos(nπ2) 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
    {12,13,14,15,... \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, ... }

    {12,25,38,411,... \frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, ... }

    {-1, 4, -9, 16, ... }

  • 3.
    Convergence and divergence of sequences

    Evaluate the limits and determine if the following limits are converging or diverging.
    lim\limn →\infty (1)nn2\frac{(-1)^n}{n^2}

    lim\limn →\infty 6(12)n6(\frac{1}{2})^n

    lim\limn →\infty n3+n+1n2+1\frac{n^3+n+1}{n^2+1}