Maths1

Országok listájaHungaryBudapesti Corvinus EgyetemGazdálkodástudományi KarNemzetközi gazdálkodás (angol nyelven)Mathematics1JegyzetekMaths1

1.1. ELEMENTS

1.1 Elements
The function

a:NR
dened on the set of natural numbers N is called (innite) sequence. We use the notation an for the n-th term. 1 Some examples: an = n, an = n , an = n+1 . n+2

Denition 1.1 The sequence an is said to be convergent and tends to A, if for any
> 0, there exists an index N , such that, |an - A| < .
whenever n N . If the sequence is convergent then A is called the limit of the sequence an and we write lim an = A .
n

If there is no such real number A, then the sequence is called divergent.

Theorem 1.1 If the sequences an and bn are convergent and limn an = A and limn bn = B then
· limn (an ± bn ) = A ± B, · limn (an · bn ) = A · B, · if B = 0 then limn
an bn

=

A B, an bn

· if A = 0 and B = 0 then limn

= ±.

1 Example 1.1 Let us consider the sequence an = n . For arbitrary > 0 let N be an

integer, greater than 1/. Then if n N

1 < , n

therefore using the 1.1 denition
n

lim

1 =0. n

Example 1.2 In a similar way we can nd the limit of other sequences. Let us
consider for example the sequence
2n2 + 5 . n2 - 6n + 8

an =

If we divide both the numerator and the denominator by n2 , then we have
an = 2 + 5/n2 , 1 - 6/n + 8/n2

where the limit of the numerator is 2 and the limit of the denominator is 1. Therefore lim an = 2 .
n

1

1.2. SEQUENCES TENDING TO INFINITY

Every irrational number can be written as a limit of a sequence of rational numbers. For example, consider the sequence a1 = 1.4, a2 = 1.41, a3 = 1.414, a4 = 1.4142 . . . than lim an = 2 n Indeed, according to the denition 1.1, if = 10-N , then |an - 2| < for n N . Typical example for a sequence which has no limit is

an = (-1)n .

1.2 Sequences tending to innity
Let us investigate the sequence

an = 2n + 5.
The terms of this sequence are greater than any given number K if n is large enough. In that case, we say, that the limit of the sequence is innity. We use the symbol to denote innity.

K there exists an index N such that for every n N we have an > K This is expressed in the formula lim an = + ,
n

Denition 1.2 We say that the sequence an approaches + if for any real number

In a completely analogous way is dened the fact that a sequence approaches -- that is limn an = -.

1.3 Squeezing theorem
Often the limit of a sequence can be determined with the aid of other sequences the limits of which are known. Such a situation is described by the Squeezing theorem.

Theorem 1.2 Let an , bn and cn be sequences such that for every index n
an bn cn

holds and, moreover, the sequences an and cn converge to the same limit A. Then the sequence bn is convergent and limn bn = A. Example 1.3 Let a > 1 be a real number and consider the sequence bn = n a. Since a > 1, the terms of the sequence can be written in the form n a = 1 + hn , where hn > 0. By the binomial theorem we get
a = (1 + hn )n > 1 + nhn .

Rearranging the inequality it follows that
0 < hn < a-1 . n

The term on the right hand side tends to zero, hence, by the Squeezing theorem hn 0, that is n a 1.
Obviously, if 0 < a 1, then we can investigate in a similar way the reciprocal of the sequence. 2

1.4. BOUNDED AND MONOTONIC SEQUENCES

1.4 Bounded and monotonic sequences
Of course, the terms of a sequence approaching innity cannot stay between two real numbers. We introduce the following denition.

Denition 1.3 The sequence an is bounded from above, if there is a real number K

such that an K for every index n. If there is a real number K such that an K for every index n, the sequence is said to be bounded from below. A sequence is called bounded if it is bounded both from above and from below.

Example 1.4 Decide whether the sequence
an = 4n2 2n +5+8

is bounded or not? Dividing both the numerator and the denominator by 2n we get
an = 1 1 + 5/4n2 + 8/2n ,

hence 0 an 1. Thus the sequence is bounded. It is also clear that the smallest upper bound of the sequence is 1, while 0 is a lower bound, but not the greatest one.

Denition 1.4 We say that the sequence an increases if an an+1 for every index n. A decreasing sequence is dened similarly. A sequence that is either increasing or decreasing is called monotonic. Example 1.5 Consider the sequence
an = 2n - 1 . n+2

We have

2n + 4 - 5 5 =2- . n+2 n+2 The value of the fraction subtracted from 2 decreases if n increases, therefore the sequence an is increasing. It is also clear that the sequence is bounded from above and its smallest upper bound is 2. Moreover, an =
n

lim an = 2 .

Our next theorem states that this property is characteristic for bounded monotonic sequences.

Theorem 1.3 An increasing sequence which is bounded from above is convergent.
An analogous statement holds for decreasing sequences that are bounded from below. The sequence n 1 an = 1 + n is of particular importance. It can be shown that this sequence is increasing and bounded from above, therefore it is convergent. The limit of this sequence is denoted by e. It can be shown that e = 2.7182... is irrational. 3



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