Task

You are going to investigate the behaviour of series of fractions and the sums of those series.
Here are a few examples:

1/1 + 1/2 + 1/3 + 1/4 ... (... means and so on.)

1/1 + 1/2 + 1/4 + 1/8 ......

1/1 + 1/(2*1) + 1/(3*2*1) + 1/(4*3*2*1) ..... ( 2*1 means 2 times 1. 4*3*2*1 is called a factorial or 4! for short.)

You are going to try to find out what happens if you carry these series on for 10, 20, 100 or a huge number of terms.

You are going to use the Excel program to help with this investigation.

Start a new workbook and number down the leftmost column.

You can save effort by selecting the first three rows and then draging down the column. This will autofill with numbers that should go up by one every time.

You now need to sort out the second column which will be the fractions column.
To do this use the formula "= 1/A2 " (press Return) in cell B2 and then copy it down the B column.

It should look like this.

It is much more fun to have this column shown as fractions so select the whole B column and click on the Format tab.

Click the cells option

Chose the Fractions and Three Digit options and click okay

Your workbook should look like this.

Now add two more columns for the running total of your fraction series. You can format one of these totals as a fraction and keep the other as a decimal.
The reason why becomes obvious soon!

To obtain the running total use this formula and copy it down the columns.

This is the formula for the decimal column.

Your workbook should look like this.

Format the decimal column to 15 decimal places and then copy the whole bottom line of the chart down for about 100 rows or so.

Does the series show any signs of reaching a final limit?

Draw a suitable graph of your results

Powers of 1/2

Use exactly the same method to investigate this series.

1/1 + 1/2 + 1/4 + 1/8 ......

This is the formula that you need in cell B2 to calculate the powers of 2.

=1/2^(A2-1)

^ is Excel for "to the power of".

Your workbook should look like this.

Investigate this series for about 50 terms.
What happens? Does it reach a limit?
Draw a graph of this series

Exponential Series

The final series is a form of the exponential series.

1/1 + 1/(2*1) + 1/(3*2*1) + 1/(4*3*2*1) ..... ( 2*1 means 2 times 1. 4*3*2*1 is called a factorial or 4! for short.)

Use exactly the same methods but use this formula to get the fraction term.

=1/FACT(A2)

Your workbook should look like this.

If you study the bottom of the chart then you should be able to see that Excel cannot cope with some of the numbers involved.

Investigate this series for about 50 terms.
What happens? Does it reach a limit?
Draw a graph of this series

Extensions and Challenges

Make up your own series that include fractions. Try to work out how to get them into Excel and then study what happens when you extend them to large numbers of terms.

Here are a few more ideas

1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6...

1/1 + 1/3 + 1/9 + 1/27 ...