What is the Limit of a Sequence?


A sequence ana_n corresponds to infinite array or list of number of the form

a1,a2,a3,....a_1, a_2, a_3, ....

where a1,a2,a3,...a_1, a_2, a_3, ... are real numbers. For example, the sequence

an=1na_n = \frac{1}{n}

is represented by the list

1,12,13,14,....1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ....

because those are the values the expression an=1na_n = \frac{1}{n} takes when nn takes the values 1, 2, 3, ...etc.

Convergence of sequences

One concept that is typically hard to grasp is the convergence of a sequence. The idea is very trivial though: A sequence ama_m converges to a value aa if the values of the sequence get closer and closer to aa (in fact they get as close as we want) as nn approaches to infinity.

For example: The sequence an=1/na_n = 1/n is such that

an=1n0a_n = \frac{1}{n} \to 0

because the value of 1/n1/n becomes "as close to zero as we want" as nn approaches to infinity.

Formal Definition of Convergence

Formal definition of convergence:

The sequence anaa_n \to a as nn \to \infty, or otherwise said limnan=a\lim_{n \to \infty}{a_n} = a if

• For all ε>0\varepsilon >0, there exist n0n_0 such that nn0      ana<εn \geq n_0 \,\,\, \Rightarrow \,\,\, |a_n - a|< \varepsilon

This is saying that no matter how close you want the sequence from aa, there is always a point in the sequence such that all the points further than that, are close enough to aa. In other words the convergence of a sequence doesn't state that some number of the sequence get close enough to the limit aa, but instead, it indicates that if we go far enough into the sequence, all the values of if will be close enough.

Algebra of Limits

Operating with limits is not as complicated once we know some them. In fact, there are simple rules that allow to compute more complicated limits based on simpler ones. These rules are shown below:

If limnan=a\displaystyle\lim_{n \to \infty}{a_n} = a and limnbn=b\displaystyle\lim_{n \to \infty}{b_n} = b then we have:

(1) limn(an+bn)=limnan+limnbn=a+b\displaystyle\lim_{n \to \infty}(a_n + b_n) = \displaystyle\lim_{n \to \infty}{a_n}+\displaystyle\lim_{n \to \infty}{b_n} = a + b

(2) limnanbn=limnan×limnbn=ab\displaystyle\lim_{n \to \infty}{a_n b_n} = \displaystyle\lim_{n \to \infty}{a_n}\times\displaystyle\lim_{n \to \infty}{b_n} = a b

(3) limnanbn=limnanlimnbn=ab\displaystyle\lim_{n \to \infty}\frac{a_n}{b_n} = \frac{\displaystyle\lim_{n \to \infty}{a_n}}{\displaystyle\lim_{n \to \infty}{b_n}} = \frac{a}{b}

(where property (3) holds as long as b0b \ne 0 .)

Example: The limit

limnn2n2+1\lim_{n \to \infty}\frac{n^2}{n^2 + 1}

is computed by first multiplying both numerator and denominator by 1n2\frac{1}{n^2}, which means

limnn2n2+1=limn11+1n2=11=1\lim_{n\to\infty}\frac{n^2}{n^2 + 1} = \lim_{n\to\infty} \frac{1}{1 + \frac{1}{n^2}}= \frac{1}{1} = 1

because limn1n2=0 \displaystyle\lim_{n\to\infty} \frac{1}{n^2} = 0.

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