Lesson 13.4: Series and Their Notations

New Concept

A series is the sum of terms in a sequence. We use summation notation, $\sum$, to represent these sums. You will learn to calculate sums for arithmetic, geometric, and even infinite series, applying these skills to solve problems like annuities.

What’s next

This is just the start. Next, you'll tackle practice cards on using summation notation and work through interactive examples of arithmetic and geometric series.

Property

The sum of the terms of a sequence is called a series. Summation notation is used to represent series and is often known as sigma notation because it uses the Greek capital letter sigma, $\Sigma$. The sum of the first $n$ terms of a series can be expressed in summation notation as follows:

This notation tells us to find the sum of $a_k$ from $k=1$ to $k=n$. $k$ is called the index of summation, $1$ is the lower limit of summation, and $n$ is the upper limit of summation. To evaluate, substitute each value of the index $k$ from the lower limit to the upper limit into the formula and add the resulting terms.

Examples

• Evaluate $\sum_{k=2}^{5} (k-1)^2$. We sum the terms for $k=2,3,4,5$: $(2-1)^2 + (3-1)^2 + (4-1)^2 + (5-1)^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1+4+9+16=30$.

• Evaluate $\sum_{i=1}^{4} (3i+2)$. We sum the terms for $i=1,2,3,4$: $(3(1)+2) + (3(2)+2) + (3(3)+2) + (3(4)+2) = 5+8+11+14=38$.

Property

An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first $n$ terms of an arithmetic sequence is:

To find the sum, identify the first term $a_1$, the last term $a_n$, and the number of terms $n$, then substitute these values into the formula.

Examples

• Find the sum of the arithmetic series $3 + 7 + 11 + 15 + 19$. Here, $a_1=3$, $a_n=19$, and $n=5$. So, $S_5 = \frac{5(3 + 19)}{2} = \frac{5(22)}{2} = 55$.

• Find the sum of the series $50 + 45 + 40 + \dots + 10$. First, find $n$: $10 = 50 + (n-1)(-5)$, which gives $n=9$. Then, $S_9 = \frac{9(50 + 10)}{2} = \frac{9(60)}{2} = 270$.

Property

A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first $n$ terms of a geometric sequence is represented as:

To use the formula, identify the first term $a_1$, the common ratio $r$, and the number of terms $n$.

Examples

• Find $S_6$ for the series $2 + 6 + 18 + \dots$. Here, $a_1=2$, $r=3$, and $n=6$. The sum is $S_6 = \frac{2(1 - 3^6)}{1 - 3} = \frac{2(1 - 729)}{-2} = -(-728) = 728$.

• Evaluate $\sum_{k=1}^{5} 100 \cdot (\frac{1}{2})^k$. This is a geometric series with $a_1 = 100(\frac{1}{2})^1=50$, $r=\frac{1}{2}$, and $n=5$. The sum is $S_5 = \frac{50(1 - (\frac{1}{2})^5)}{1 - \frac{1}{2}} = \frac{50(1 - \frac{1}{32})}{\frac{1}{2}} = 100(\frac{31}{32}) = 96.875$.

Property

The sum of an infinite series is defined if the series is geometric and the common ratio $r$ satisfies $-1 < r < 1$. The formula for the sum of an infinite geometric series is:

To find the sum, identify the first term $a_1$ and common ratio $r$, confirm that $-1 < r < 1$, and then use the formula.

Examples

• Find the sum of the infinite series $18 + 6 + 2 + \dots$. The first term is $a_1=18$ and the common ratio is $r = \frac{6}{18} = \frac{1}{3}$. Since $-1 < \frac{1}{3} < 1$, the sum exists. $S = \frac{18}{1 - \frac{1}{3}} = \frac{18}{\frac{2}{3}} = 27$.

• Determine if the sum of $2 - 4 + 8 - \dots$ is defined. The common ratio is $r = \frac{-4}{2} = -2$. Since $r$ is not between -1 and 1, the series diverges and its sum is not defined.

Property

An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. The future value of an annuity can be found using the formula for the sum of a geometric series:

To find the value: determine $a_1$ (the periodic deposit), $n$ (the number of deposits), and $r$ (1 + the periodic interest rate). Then substitute these values into the formula.

Examples

• A deposit of 300 dollars is made monthly into a fund earning 3% annual interest, compounded monthly, for 10 years. We have $a_1=300$, $n=120$, and $r = 1 + \frac{0.03}{12} = 1.0025$. The value is $S_{120} = \frac{300(1 - 1.0025^{120})}{1 - 1.0025} \approx 41822.44$ dollars.

• A person deposits 1000 dollars every quarter for 5 years. The account earns 8% annual interest, compounded quarterly. Here $a_1=1000$, $n=20$, and $r = 1 + \frac{0.08}{4} = 1.02$. The value is $S_{20} = \frac{1000(1 - 1.02^{20})}{1 - 1.02} \approx 24297.37$ dollars.