5.1 Introduction to sequences

Introduction to sequences

Lessons

Notes:
Note:
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:



Where the sequence {xnx^n} is convergent for -1< xx \leq 1, and divergent if xx > 1.
  • 1.
    Overview:
  • 2.
    Finding the terms of a sequence

    Find the first five terms of the following sequences.
  • 3.
    Finding the formula for a sequence

    Find the formula for the general term ana_n for the following sequences
  • 4.
    Convergence and divergence of sequences

    Evaluate the limits and determine if the following limits are converging or diverging.
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Introduction to sequences

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