When we add the terms of a sequence together, we form a series.
We use the symbol Sn to show the sum of the first n terms of a series.
So, Sn = T1+T2+T3+T4+…+Tn
Arithmetic Series
The formula is
Where:
Sn is the sum of n terms.
a is the first term.
n is the number of terms.
d is the common difference.
A. Proof
The general term of an arithmetic series is Tn=a+(n-1)(d).
So, Sn = T1+T2+T3+T4+…+Tn
Example 7:
a) Determine the sum of the first 20 terms of the series: 3 + 7 + 11 + 15 + …
b) The sum of the series 5 + 3 + 1 + … is –216. Determine the number of terms in the series.
Solutions:
Geometric series
The formula for a geometric series is:
Where:
a is the first term.
r is the common ratio.
n is the number of terms.
Sn is the sum of the terms.
Example 8:
Evaluate 25 + 50 + 100 + … to 6 terms.
Solution:
We need to check if this is an arithmetic series (common difference) or a geometric series (common ratio) first.
You should see that there is a common ratio of 2 because 50⁄2=2 and 100⁄50; therefore, r=2.
So, the sum of the first 6 terms of the series is 1 575.
Sigma Notation
Here is another useful way of representing a series. The sum of a series can be written in sigma notation.
The symbol sigma is a Greek letter that stands for ‘the sum of’.
Example 9:
Convergence of Sequences and Infinite Geometric Series
An infinite series is one in which there is no last term, meaning that the series goes on without ending. Understanding the behaviour of sequences and series as they progress towards infinity is a cornerstone of mathematical analysis, with applications ranging from calculus to financial mathematics.
A. Notation
The notation of convergent and divergent sequences and series follows mathematical conventions.
Sequences are denoted by , where indicates the term’s position. The limit of a sequence as approaches infinity is .
For series, the sum is indicated by the symbol , with the infinite geometric series , identifying as the first term and as the common ratio.
B. Divergent and Convergent Series
The sum of divergent series progress towards infinity, while the sum of convergent series approaches a finite value
Example 10:
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 which approaches infinity.
Example 11:
Consider the infinite series:
This series will converge to 2. It is therefore called a convergent series and we can write the sum to infinity equals 2: S∞=2
C. Investigating and Interpreting Convergent Series
You can identify a convergent infinite series by looking at the value r.
An infinite series is convergent if -1< r <1, r≠0
The formula for the sum of a convergent infinite series is given as:
Where a is the first term, r is the common ratio.
Convergence of a sequence {an} to a limit L is defined by the ability to find, for every value of n, a natural number N. This is denoted as lim┬(n→∞) (an)= L.
In the context of an infinite geometric series, convergence can also be expressed as the sum of the individual terms up to infinity: ∑∞n=1 arn-1
Convergence can be identified by evaluating the value r, known as the common ratio. The series converges if the absolute value of the common ratio is less than one (|r| < 1), with its sum expressed as S= a ÷ (1-r).
Here, a is the first term of the series. To determine the value of r, the ratio of successive terms in the series is calculated through the equation r = an+1 ÷ an , where an and an+1 represent successive terms in the series. This ratio must remain constant for the series to be geometric and must satisfy |r| < 1 for convergence.
For example, the value of r for the infinite series in Example 11 is calculated as follows: r= an+1 ÷ an = 0.5÷1 = 0.25 ÷ 0.5 = 0.5
Therefore, the infinite series will converge as r satisfies both criteria:
- |r| < 1
- r remains constant.
The concept of convergence for sequences implies that the sequence’s terms grow increasingly close to a specific limit value as n becomes very large. For an infinite geometric series, convergence signifies that the series’ sum approaches a finite value as more terms are added. The determination of r is crucial here; it dictates whether the series converges or diverges. A series is divergent if the criteria for convergence are not met, highlighting the importance of correctly determining and interpreting the value of r in the context of infinite geometric series.