Notes on using GPSum.xls

This spreadsheet allows students to explore the behaviour of the sum of the terms of geometric series as the number of terms increases.

Background:

Students will need to understand the terms convergent, divergent, periodic, oscillating  and limiting value with respect to the behaviour of a sequence.

It is assumed that they have covered the formula for the sum to n terms of a Geometric series.

Questions for students.

• Experiment with changing the values of a and r in the spreadsheet. List the types of behaviour that are possible (e.g. oscillating and convergent, or oscillating and divergent)?

• For each type of behaviour try to find out exactly which values of a and r will produce it so that if someone gives you a pair of values or a and r you can predict the behaviour. It is a good idea to be systematic in the way that you vary the values and to record your evidence in a table like the one shown.

Value of a	Value of r	Type of behaviour	Limiting value if it exists.
1	2	Divergent	-
1	-0.5	Oscillating and convergent.	0.66666
1	0.5	Convergent	2

• For the convergent cases try to find a formula for the limiting value in terms of a and r. State the values of a and r that your formula is valid for.

• Pair up with another student and test each other. Give a pair of values of a and r to your partner and see if they can predict the behaviour and limiting value. Use the spreadsheet to check.

• Use the formula for the sum to n terms of a Geometric series to try to prove your formula for the limiting value.