25.5 Infinite geometric series
When you count by 5, 6, 7, 8, 9 and onwards you would realize that you’re actually counting an ordered list of numbers. Ordered list of numbers are otherwise referred to as sequences. Consider we’re given the sequence 11, 9, 7, 5, 3, 1… If you would see, there are three dots ending the list of numbers. This indicates that this sequence is infinite and it can go on as long it could.
There are two kinds of sequences, infinite, and finite. We already saw an example of infinitesequence in the first paragraph and from there we know that they can go on and on. For the finite sequence they can just be limited with a few terms. Terms or Elements are the numbers in a sequence. The sequence has a common difference and a common ratio. This would help establish the relationship between the successive terms in the sequence. When we determine the sum of a sequence then we are able to find the series.
Apart from looking at the regular numbers like 1,2,3,4,5 and so on, we will be able to study other expressions which might be consisted of a coefficient, a variable and an exponent and other expressions such as factorials. In this chapter, we will also learn how to use mathematical induction to prove the sum of the series.
Infinite geometric series
Basic concepts:
 Arithmetic series
 Geometric sequences
 Geometric series
Lessons
Notes:
if $1 < r < 1$, an infinite series is convergent: S_{ ∞}= $\frac{t_1}{1r}$
Otherwise, an infinite series is divergent: S_{ ∞}= undefined

1.
Using the common ratio to determine whether a sum to infinity exists
For each geometric series determine the:
i) common ratio.
ii) sum of the first 10 terms.
iii) sum to infinity.

2.
Expressing a repeating decimal as an infinite geometric series
For the repeating decimal:
i) express it as an infinite geometric series.
ii) write it as a fraction by evaluating the sum of the infinite geometric series.