What are Infinite series? An infinite series is the sum of an infinite number of terms that follow a specific rule.
How to use infinite series:
An infinite series adds up all the terms of an infinite sequence and follows a set rule.
For example in the sequence 20, 40, 60….., the rule is n+20. 20 is added to each term to get the next term.
The infinite series would be 20+40+60+...
The dots at the end tell us that the sequence is infinite and will continue to keep going.
Convergent vs Divergent
An infinite series is said to be convergent if the sum of all the terms approaches a finite number.
For example:
1/3+1/9+1/27+1/81….
This sequence is adding up to get closer and closer to 1.
An infinite series is said to be divergent if the sum of all the terms does not approach a finite number.
For example:
10,20,30,40…..
This sequence will not approach a finite number. It will approach infinity.
Geometric Series vs Arithmetic Series:
An arithmetic sequence is a sequence in which a constant is added or subtracted to each term in order to get the next term.
For example in the infinite sequence 2,5,8,11…., each term has the constant 3 added to it in order to get to the next term.
The formula to find the nth term in an arithmetic series is:
Where,
d = the common difference between each term.
Example:
The arithmetic sequence is
81, 73, 65, 57,...
Find the value of a₃₁.
In this example,
d = -8
n=31
Using the formula:
We get:
Therefore, a₃₁= -159
A geometric sequence is a sequence in which each term is multiplied by a constant in order to get the next term.
For example in the infinite sequence 10,50,250,1,250…., each term is multiplied by the constant 5 in order to get to the next term.
The formula to find the nth term in a geometric series is:
Where,
a= the first term of the sequence and
r= the common ratio between each term.
Example:
Write the equation for the nth term of the geometric series and find the value of the fourth term
-6,-42,-294,...
In this example,
a= -6
r= 7
n=4
Using the formula:
We get:
Therefore the fourth term is -2,058.
Sample Math Problems
Question
Find the next two numbers from the arithmetic sequence 3, 3.5, 4, 4.5…..
Solution
In this problem we first need to look at what is added or subtracted to each term to find the next two terms.
Looking at the sequence, we see that the common constant added to each term is 0.5.
To find the next term, we must add 0.5 to the last term 4.5.
4.5+0.5= 5
Now to find the 6th term, we need to add 0.5 again to 5.
0.5+5=5.5
Therefore, the next two terms are 5 and 5.5.
Question
Write the equation for the nth term and find the value of a₂₅
Solution
The arithmetic sequence is 5,8,11,14….
To solve this problem use the formula:
