Sum of a Finite Series

Our text gives examples of BASIC language programs that can be used on a computer to find the sum of a series. This works, but spreadsheets are far faster and more efficient in many cases.

Example 1 - Page 486

This example asks you to "Find the sum of the cubes of the first twenty positive integers." and demonstrates a BASIC computer program to solve the problem. Here is a spreadsheet solution (The diagram at right shows the formulas in the spreadsheet): 1. (Optional) In cell A1, type the problem name.
2. (Optional) in cell A2 and A3, type a description of the problem.
3. In cell A5, enter the column label "n", the term number.
4. In cell A6, enter "1" for the first term number.
5. In cell A7, enter "=A6+1" for the second term number.
6. In cell B5, enter the label "Sn" for the sum of n terms of the series.
7. In cell B6, enter "=A6^3" for the sum for one term.
8. In cell B7, enter "=B6 + A7^3" for the sum for two terms.
9. Click on cell A7, then drag the mouse across to cell B7, then drag the mouse down to (at least) cell B25. Select "Fill Down" from the Edit menu to fill in the rest of the spreadsheet.
10. The answer is in cell B25. The finished spreadsheet is shown at left. Notice that the column headings in row 4 have been aligned to the right (Format Menu- Align - Right).

Modifying the Spreadsheet to Solve Another Problem

To find the sum of some other sequence, you could modify this spreadsheet by changing column B. Change cell B6 to the first term of the series, and enter the recursive definition of the series in cell B7.

An Alternate Solution

Here is a spreadsheet that shows the terms of the series, as well as its sum: 1. (Optional) In cell A1, type the problem name.
2. (Optional) in cell A2, type a description of the problem.
3. In cell A4, enter the column label "n", the term number.
4. In cell A5, enter "1" for the first term number.
5. In cell A6, enter "=A6+1" for the second term number.
6. In cell B4, enter the label "Tn = n^3" to label the terms column.
7. In cell B5, enter "=A5^3" for the first term.
8. In cell B6, enter "=A6^3" for the second term.
9. In cell C4, enter the label "Sn" for the sum of n terms of the series.
10. In cell C5, enter "=B5" for the sum for one term.
11. In cell C6, enter "=C5+B6" for the sum for two terms.
12. Click on cell A6, then drag the mouse across to cellC6, then drag the mouse down to (at least) cell C24. Select "Fill Down" from the Edit menu to fill in the rest of the spreadsheet.
13. The answer is in cell C24. The spreadsheet on the left shows the finished product. Note that the column headings have been aligned, and commas have been added to the numbers (Format Cells - Number - Use 1000 separator).

Modifying this Spreadsheet to Solve Another Problem

To find the sum of some other sequence, you could modify this spreadsheet by changing column B. Change cell B5 to the first term of the related sequence, and enter the recursive definition of the sequence in cell B6.

last update February 2, 2008 by JL Stanbrough