**4.2.1. Sums, series and Sigma summation notation**

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

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

Our recommendation is to:

**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.**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.**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.**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,

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

\( a_k = a_1 + (k-1)d \),

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

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.

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

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.