4.2. Sequences & Series: Sums & series

Learning Objectives

You should be able to

• Understand notation and terminology regarding series
• Use sigma summation notation to describe/understand a sum or series
• Identify when a sum/series is arithmetic or geometric
• Write down a formula for the a finite sum of an arithmetic or geometric sum term of an arithmetic or geometric progression

Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.
Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

Tutorials: How to guides

4.2.1. Sums, series and Sigma summation notation

$$\displaystyle \sum_{k=n}^N a_k= a_n+a_{n+1}+\dotsb +a_N$$

4.2.2. Arithmetic Series

$$a_k = a_1 + (k-1)d$$,
$$\displaystyle \sum_{k=1}^n a_k = a_1 + (a_1+d) + (a_1 + 2d) + \dotsb +(a_1+(n-1)d) = n\left(\frac{a_1+a_n}{2}\right)$$

4.2.3. Geometric Series

$$\displaystyle \sum_{k=0}^{n-1} ar^k= a+ar+ar^2 + \dotsb+ar^{n-1}= a\frac{1-r^n}{1-r}$$

Economic Application: Discounting and bond pricing

This application formulates the important economic principle of the time value of money and explains its implications in the context of valuing streams of payments with different payments accruing at different points in time. The idea of discounting is then applied to the context of finding the fair price of types bonds, known as annuities and perpetuities, by using the formulas for geometric series.

Economic Application Exercise

The following quiz allows you to test your understanding of finding the fair prices of annuities and perpetuities.

Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to If you want to test yourself further, then try the advanced quiz linked below.