Maths: Pi from a spreadsheet

That is, π from a spreadsheet. This is heavy grade maths, under 15s should avoid it, unless they are very numerate.

There are quite a few series that converge on a value of π, or some function of π. Here’s one:

(π⁴)/96 = 1/1⁴ + 1/3⁴ + 1/5⁴ + 1/7⁴ + 1/9⁴ +…

These instructions will help you to create a spreadsheet that will get to π to about five decimal places in about 250 rows.

Begin with the value 1 in cell A2

Now from the home tab, select FILL SERIES (step 2) to get the numbers 3, 5, 7 … 199 in the cells down to A101. Column A is now ready.

Now enter this formula in B2 : =1/(A2*A2*A2*A2)

And put this in C2: =C1 + B2 (this will give us a running total of column B, up to that row).

Then put this in D2: =SQRT(SQRT(C2*96)) to get your first estimate of the value of π.

Now you can highlight cells B2, C2 and D2, and then highlight down to row 250, and use FILL DOWN from the Home tab to copy the formula down into those rows as well, and extend column A down to row 250 (think about this!).

Check the answers that you get in cells D246 and D247, after you have extended the spreadsheet down to row 250. The value you should be aiming at is 3.141 592 653 589 793 238 462… but getting there will take a while longer…J

If anything does not work, check the values in column A, which has to contain only the consecutive odd numbers. Then work your way across, checking each of the formulas in the instructions, until you spot the mistake you made. The formulas given here worked (many years ago) in MS Works and they still work in MS Excel (2010 version), but they have not been tested on other or more recent spreadsheets.

Here is what you will see: code on the left, output on the right.

Here, without explanation, are two other convergent series that close in on π.

π²/₆=1/1²+1/2²+1/3²+1/4²+1/5²+…

π/4=1-¹/₃+¹/₅-¹/₇+¹/₉…

There are more of these: use <infinite series pi> as your web search string.

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