Limit of a Sequence

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Often we are interested in value that sequence will take as number n becomes very large.

Definition. Constant number a is called a limit of the sequence x_n if for every epsilon>0 there exists number N, such that all values x_n whose number n>N, satisfy inequality |x_n-a|<epsilon.

The fact that a is a limit of sequence is written as limx_n=a or limx=a or lim_(n->oo)x_n=a.

Also we say that variable approaches a: x_n->a or x->a.

This definition can be reformulated in simple words: a is a limit of the sequence if diffrerence between its values and a becomes very small (|x_n|-a<epsilon), starting with some number N.

Definition. Sequence is called convergent (converges to a) if there exists such finite number a that lim_(n->oo)x_n=a. Otherwise, sequence diverges or divergent.

In other words sequence is convergent if it approaches some finite number.

Note, that symbol oo means infinity (very large number). So, if sequence has limit a then members in this sequence starting with some number N approach a.

For example, sequence x_n=1/n and corresponding list {1,1/2,1/3,1/4,1/5,...} approach 0 because x_(10000)=1/(10000) and x_(100000)=1/(100000); we see that the greater n, the closer value to 0.

It is important to notice, that number N depends on the choice of epsilon. Therefore, we will write sometimes N_(epsilon) instead of N. In general the lesser epsilon, the greater N: if we require more closeness, the more bigger values of sequence we need to consider. The only exception is when all values in sequence are same and equal a. In this case a=lim x_n, and inequality will hold for any epsilon for all x_n (same can be said if values of sequence equal a, starting from some number).

By Property 1 of absolute values |x_n-a|<epsilon is equivalent to -epsilon<x_n-a<epsilon

or a-epsilon<x_n<a+epsilon.

Therefore, we have geometric interpretation of limit. If we take any segment of length 2epsilon with center a, then all points x_n starting with some number should lie within this segment (only finite number of points can lie outside the interval).

Also sequences can approach same value but approach it in different manner.

Consider two sequences: x_n=1-1/n (corresponding list of members is {0,1/2,2/3,3/4,4/5,...} )and y_n=1+2 ((-1)^n)/n (corresponding list of members is {-1,2,1/3,3/2,...}).

See how, closer becomes point to line y=1 when n becomes larger. Thus, it is natural to suggest that both sequences have limit 1. But they approach it differently: if we on x-axis set natural numbers and on y-axis values that sequence will take then we will obtain following graph (see figure below). But we are not interested how sequence behaves from the "start", we are interested how it behaves when n becomes very large.

Example 1. Show that limit of x_n=1/n+1 equals 1.

To prove this by definition we need for every epsilon>0 find such natural number N that for every n>N |x_n-1|<epsilon.

So, |x_n-1|=|1/n+1-1|=|1/n|.

Thus, |1/n|<epsilon when n>1/epsilon. Therefore, we can take N_epsilon=[1/epsilon], where [x] is a floor function.

If we take epsilon=0.01 then N_epsilon=[1/0.01]=100. So, members starting with 101-th number will be different from 1 on less than epsilon: Indeed, x_(101)=1/101+1=1.0099 and |1.0099-1|<0.01.

If we take epsilon=0.5 then N_epsilon=[1/0.5]=2. So, members starting with 3-rd number will be different from 1 on less than epsilon: Indeed, x_(3)=1/3+1=4/3~~1.3333 and |1.3333-1|<0.5.

As can be seen for different epsilon there are different values of N_epsilon. Limit exists when for any epsilon we can find corresponding N_epsilon.

Therefore, lim_(n->oo)x_n=lim_(n->oo)(1+1/n)=1.

Example 2. Show that sequence x_n=(-1)^n doesn't have limit.

This sequence is represented by list {-1,1,-1,1,-1,1,...}.

If we take epsilon=0.01 then we can't find N such that for n>N members will be close to some number (limit), because members oscillate: sequence takes by turn values 1 or -1. Thus, this sequence doesn't have a limit.

Following fact closes this note.

Fact. Sequence can't have more than one limit.

This means that sequence either doesn't have limit or has exactly one limit.