# 3. Limit of a sequence

Part 1.: Exercises, solutions.
Part 2.: Exercises, solutions.

### Definition

The notion of sequence convergence seems intuitively clear. A sequence converges to limit if its elements are closer and closer to . E.g. sequence , for converges to zero. Speaking more precisely, for any , there exists , such that for all , distance between and is less than . So:

And indeed, , because if , let , then, if , then .

Meanwhile, sequence , does not converge to any limit, since for any and , for all , or .

### Squeeze theorem

Given two sequences i , which have limits and respectively and such that for any we have , we have also that .

Therefore if we study convergence of a sequence we may try to find sequences and such that for all and . Then also converges to — this theorem is usually called the squeeze theorem.

E.g, let , . Then for all . Since , also .

### Arithmetic of limits

We have a natural arithmetical theorem on limits. If and , then:

• ,
• ,
• ,
• , if (assuming that for any

Let us calculate the limit of . Notice that:

Since , , the numerator converges to , and since and , the limit of denominator is (we apply the theorem on arithmetic of limits). Therefore (again we apply the theorem).

### Bounded sequences

A sequence is bounded, if there exists , such that for any . We have the following theorem: if converges to zero and is bounded, then also converges to zero.

E.g. let . Obviously , where is bounded, and converges to zero. So .

### Infinite limits

If is such that for all there exists , such that for all we have , so if for arbitrarily large number from a given place on the elements of sequence are greater then this number we say that the sequence tends to infinity and write .

If a sequence is such that for any there exists , such that for all we have , we say that the sequence tends to minus infinity and write .

E.g. let . This sequence tends to infinity, because if , , let . Then for all , we get .

### Arithmetic of infinite limits

Let and be sequences of real numbers. Then

• if jest converges to any finite number or , and , then .
• if converges to any finite number or , and , then .
• if converges to any finite number g>0 or , and , then .
• if converges to any finite number g<0 or , and , then .
• if converges to any finite number g>0 or , and , then .
• if converges to any finite number g<0 or , and , then .
• if converges to any finite number g, and , then .

Notice that this theorem does not tell anything about some types of limit operation, in which we can tell nothing about the limits. E.g. operations like , or . In this cases usually some further calculations are needed to be able to use the above theorem.

E.g., let . Therefore . Since , . Because , we get that .

### Number e

Imagine that you have a bank account with interest rate of a year! But you have only in this account. If interest rates are calculated every year, after one year you will have dollars. But if the interest rates were calculated twice a year you will get a half of your money twice a year, so at the end of the year you will have . If it would be calculated four times a year, you will get , monthly . It is clear that the final amount of money increases if the frequency of calculating interest rate increases. How big this amount can be?

Obviously we ask about the limit of sequence , . The answer is quite surprising. The limit of this sequence is an irrational number, which plays a huge role in mathematics and is denoted by . .

Knowing that , we can calculate limits of many other sequences, e.g. . Notice, that , so .

### Cauchy condition

A sequence is a Cauchy sequence if for any positive arbitrarily small from some place on the distance between any two elements of the sequence is less than . So:

E.g.: sequence is a Cauchy sequence. Indeed, let . Then let . Then for all we get:

It turns out that a sequence of real numbers is a Cauchy sequence if and only if it converges to a finite number.