The purpose of this section is to discuss sums that contains infinitely many terms
For example When we write in the decimal form We mean which suggests that the decimal representation of can be viewed as a sum of many real numbers. The most familiar example of such sums occur in the decimal representation of real numbers.
The numbers are called the term of the series An infinite series is an expression that can be written in the form Definition (1) and is called the nth partial sum of series and the sequence is called the sequence of partial sums
Definition (2) be the sequence of partial sums of series Let The series is said to converge to a numberiff In which case we call the sum of the series and write If no such limit exists the series is said to diverge
Example 1 Determine whether the series converge or diverge.If it is converge, find the sum Solution: and so on the sequence of partial sums is Since this is divergent sequence and so the given series diverges and consequently no sum
Definition (3) A series of the form is called a geometric series Here are some examples
Theorem 1 A geometric series Converges ifand diverge if the sum is Proof: First, ifthen the series is If the series converges,then
Also, if then the series is the sequence of partial sums is Which diverges Moreover, Similarly for
Example 2 Determine whether the following series converge or diverge.If it is converge, find the sum a- b- c- Solution: is a converge geometric series with and the sum is
b- the series diverges c- the series is geometric series with Ifthe series is converges andthen and diverges otherwise Exercise: Find the rational number represented by the repeating decimal
Evaluate Solution: Creative thinking
Worksheet Find the value to which each the following series converges1- a-d- c-b- e- f-g- 2- A ball is dropped from height of 10m.Each time it strikes the ground it bounces vertically to height that is of the preceding height. Find the total distance the ball will travel if it is assumed to bounce infinitely often. Answer=90m