Introduction to Calculus and Analysis - Sequences

Numerous cases occur in mathematics in which a quantity α depends on a positive integer n. Such a function associates a value with every natural n. The function α(n) associates a value with every natural number n. The function α(n) is called a sequence, specifically, an infinite sequence, if n ranges over all positive integers. Usually, we write  instead of α(n) for the "nth element" of the sequence, and think of the elements forming a sequence arranged in order of increasing subscripts n;

Here the dependence of the numbers    on n may be defined by any law whatsoever, and, in particular, the values  need not all be distinct from each other. The idea of a sequence will most easily be grasped by examples.

1. The sum of the first n integers         

     S(n) = 1+2+3+...+n=(1/2)n(n+1)

is a function of n, giving rise to the sequence

     S(1)=1, S(2)=3, S(3)=6, S(4)=10, S(5)=15,  ....

2. Another simple function of n is the expression "n-factorial" the product of the first n integers.

n! = 1*2*3* ... *n.

3. Every integer n>3 which is not a prime number is divisible by more than two positive integers. whereas the prime numbers are divisible only by themselves and by 1. We can obviously consider the number T(n) of divisors of n as a function of  n itself.  For the first few numbers it is given by the table.

n    =  1 2 3  4  5  6  7  8  9 10  11 12

T(n)= 1 2 2  3  2  4  2  4  3   4    2   6

4. A Sequence of great importance in the Theory of Numbers is π(n), The number of primes less than the number n. Its detailed investigation is one of the most fascinating problems. The principle result is: The number π(n) is given asymptotically, for large values of n, by the function n/log(n),  where by log n we mean the logarithm to the "natural base" e.  The Natural constant e is about 2.17281728.

来源<<微积分和数学分析引论>>第一卷  Richard Courant and Fritz John, 1965 by Interscience Publishers, a division of John Wiley and Sons, Inc.

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