Infinite Series
Thus far we have been working only with finite sums, meaning that whenever we determined the sum of a series, we only considered the sum of the first n terms. In this section, we consider what happens when we add infinitely many terms together. You might think that this is a silly question – surely the answer will be ∞ when one sums infinitely many numbers, no matter how small they are? The surprising answer is that while in some cases one will reach ∞ (like when you try to add all the positive integers together), there are some cases one will get a finite answer. If you don’t believe this, try doing the following sum, a geometric series, on your calculator or computer:
You might think that if you keep adding more and more terms you will eventually get larger and larger numbers, but in fact you won’t even get past 1 – try it and see for yourself!
We denote the sum of an infinite number of terms of a sequence by
When we sum the terms of a series, and the answer we get after each summation gets closer and closer to some number, we say that the series converges. If a series does not converge, then we say that it diverges.
Infinite Geometric Series
There is a simple test for knowing instantly which geometric series converges and which diverges. When r, the common ratio, is strictly between -1 and 1, i.e. -1<r<1, the infinite series will converge, otherwise it will diverge. There is also a formula for working out the value to which the series converges. Lets start off with formula (vii) for the finite geometric series:
Now we will investigate the behaviour of rn for -1<r<1 as n becomes larger. Take r=½:
n=2⇒r2=½2=½⋅½=¼<½ n=3⇒r3=½3=½⋅½⋅½=⅛<¼
Since r is in the range -1<r<1, we see that rn gets closer to 0 as n gets larger.
Therefore,
The sum of an infinite geometric series is given by the formula
where a1 is the first term of the series and r is the common ratio.
For an infinite series the index of summation i takes the values 1, 2, 3, and so on, without end. To indicate that the values for i keep increasing without bound, we say that i takes the values from 1 through ∞ (infinity). Note that the symbol “∞” does not represent a number. Using the ∞ symbol, we can write the indicated sum of an infinite geometric series (with |r|<1) by using summation notation as follows:
Example 1: Sum of an infinite geometric series
Find the value of the sum
Solution:
This series is an infinite geometric series with first term 8 and ratio ¾. So
In the content of Using Sigma Notation to represent Finite Geometric Series, we used sigma notation to represent finite series. You can also use sigma notation to represent infinite series. An infinity symbol ∞ is placed above the Σ to indicate that a series is infinite.
Example 2: Infinite Series in Sigma Notation
Evaluate ∑∞n=1 24(-⅕)n-1
In this infinite geometric series, a1=24 and r=-⅕.
Sum formula
Infinite geometric series
An infinite series is one in which there is no last term, i.e. the series goes on without ending.
Ex3. S∞ =∑∞k=1 2⋅3k-1 =2+6+18+54+⋯ the sum from term 1 to infinity of (2⋅3k-1)
a2=2⋅31=6
a3=2⋅32=18
a4=2⋅33=54
⋮
The terms of this series are all positive numbers and the sum will get bigger and bigger without any end. This is called a divergent series.
Ex4 to Ex6. Find the sum of each infinite series, if it exists.
Ex4. ∑∞k=1 5⋅4k-1
Solution: Since |r|=|4|>1, the series is diverges and the sum does not exists.
Answer:
No sum exists.
Ex5. Calculate S∞ if ∑∞p=1 8⋅41-p
Solution:
To find r, find the common ratio using a1 and a2, a2 and a3.
When dividing by a fraction, you can multiply by the inverse.
Ex6. Find ∑∞n=1 ⅓n .
Solution:
This is an infinite geometric series with a1=⅓ and r=⅓.
As n approaches infinity, ⅓n approaches 0. Therefore,
Answer:
Ex7 to Ex10. If possible, find the sum of each infinite geometric series.
Ex7. ∑∞n=1 6⋅(-0.4)n-1
Solution:
The common ratio is |-0.4|<1. Therefore, this infinite geometric series has a sum. Find a1.
=6
Use the formula for the sum of an infinite geometric series to find the sum.
Therefore, the sum of the series is 30/7.
Ex8. ∑∞n=1 40⋅⅗n-1
Solution:
The common ratio is |⅗|<1. Therefore, this infinite geometric series has a sum. Find a1.
=40
Use the formula for the sum of an infinite geometric series to find the sum.
Therefore, the sum of the series is 100.
Ex9. ∑∞n=1 ½⋅⅜n-1
Solution:
The common ratio is |⅜|<1. Therefore, this infinite geometric series has a sum. Find a1.
=½
Use the formula for the sum of an infinite geometric series to find the sum.
Therefore, the sum of the series is ⅘.
Ex10. ∑∞n=1 35⋅(-¾)n-1
Solution:
The common ratio is |-¾|<1. Therefore, this infinite geometric series has a sum. Find a1.
=35
Use the formula for the sum of an infinite geometric series to find the sum.
Therefore, the sum of the series is 20.
Ex11. ▼
(i) Find the sum to infinity of the geometric series: 1+⅖+⅖2+⅖3+⋯.
(ii) Evaluate
in terms of x, where x>2.
Solution:
(i) 1+⅖+⅖2+⅖3+⋯.
This is an infinite geometric series with first term a=1 and common ratio r=⅖
This is an infinite geometric series with first term a=1 and common ratio
Evaluating pi (π) and Euler’s number (e) with series
In this unit we see how finite and infinite series are obtained from finite and infinite sequences. We explain how the partial sums of an infinite series form a new sequence, and that the limit of this new sequence (if it exists) defines the sum of the series. We also consider two specific examples of infinite series that sum to e and π respectively.
In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.
After reading Mathlibra’s pages of Sequences and Series, you should be able to:
• recognise the difference between a sequence and a series;
• write down the sequence of partial sums of an infinite series;
• determine (in simple cases) whether an infinite series has a sum.
