Merced College; Don Power

MATH 4B - Lab 5 – USING THE COMPUTER FOR SERIES. NAME ________________

The purpose of this lab is to use a computer to determine the value to which a convergent series converges.

Part 1: Rapidly converging series can be investigated productively with a spreadsheet, such as Excel.

Example for part 1: Determine the sum of the series using Excel: Enter the following:

n	term	partial sum
	12^n/(n^1.5)!
0	=12^A3/FACT(A3^1.5)	=B3
=A3+1		=B4+C3

The result, after dragging the formulas down to row 12, is

n	term	partial sum
	12^n/(n^1.5)!
0	1	1.0000000000000000
1	12	13.0000000000000000
2	72	85.0000000000000000
3	14.4	99.4000000000000000
4	0.51428571	99.9142857142857000
5	0.00623377	99.9205194805195000
6	3.4251E-05	99.9205537319823000
7	5.5966E-09	Conclusion: Since the sequence of partial sums converges to 99.9205537375793000, this is the sum of the series 99.9205537375790000
8	3.8255E-13	99.9205537375793000
9	4.7386E-19	99.9205537375793000
10	7.5299E-24	99.9205537375793000
11	1.9974E-30	99.9205537375793000
12	2.6653E-37	99.9205537375793000

Note: An easy way to change a column width is to place the cursor at the extreme top right edge of the column to be changed (between the column letters), so that the cursor becomes a two-headed arrow. Click and hold with the mouse and drag to the side. To increase the accuracy of the partial sum beyond what Excel starts with, highlight the numbers in that column and use the toolbar button to "Increase Decimal."

Exercise for part 1:

1. Use the technique above to find the value to which converges.

2. After doing your calculations, type your name on the worksheet.

3. Print the worksheet.

4. Look up this series in our textbook (Use the index: You're looking for a table that lists several Maclaurin series). Remember that the “x” in the textbook is replaced by a number in our series.

What function does the series represent? ___________________

At what “x” is that function is being evaluated? ___________________

Evaluate the function at this “x” with your calculator. It should match your Excel value.

Part 2. Some functions converge more slowly. If the related continuous function can be integrated, the sum S can be computed to any desired level of accuracy (in theory) by using the inequality (the only difference between the expressions on the left and right is the lower limit of integration)

S_n + £ S £ S_n +

To get k (for example 6) decimal positions of accuracy in S, calculate enough terms so that left and right sides of the inequality are the same through the kth (6th, for the example) decimal position.

Example for Part 2: Determine the sum of the series using Excel. Note that before you start with Excel, you must calculate the integrals and yourself.

Once the integrals are calculated, open an Excel worksheet and make the following entries:

Excel uses "ATAN" for the arctangent function. In general, you may have to research Excel’s formulas for the function you need. As soon as you type the initial = for a formula, the symbol f_x appears on the formula bar. Click on it and you can browse through the index of functions.

The result, after dragging the formulas down to row 10000 (I’ve hidden most of the rows using Format...Row...Hide), is

Notice that n=11 is the first row that guarantees 2 decimal places of accuracy (The two columns on the right both round to 0.91 at n=11 but not before).

Also notice that the final two columns give us a better estimate of the infinite sum than the partial sum does. Would it make sense to estimate the true value of the integral as the midpoint of the last two columns? (Compare this value with the Maple result in part 3 below).

Exercise for Part 2.

1. Use this technique to find the sum of the series . You should do the integration for by using the integration tables in the back of the textbook, and check it with Maple. Which integration formula applies? Formula Number: ____________

2. Drag your formulas far enough into the series to guarantee an accuracy of 5 decimal places.

Which is the first n for which you can guarantee accuracy to the 5th decimal place? ________

Which is the first n for which you can guarantee accuracy to the 4th decimal place? ________

Which is the first n for which you can guarantee accuracy to the 3rd decimal place? ________

3. In your Excel worksheet, hide all the rows after n=10 except for these three rows (the ones first showing 3-place, 4-place and 5-place accuracy): highlight the rows to be hidden and select Format...Rows...Hide.

4. Type your name on the worksheet and print it.

Part 3: You can also investigate series using Maple’s Sum command.

Type the expression to be summed.

Highlight the output, right-click on the highlighted expression, and ask Maple to construct a sum (select Constructions...Sum).

Type the first and last indices into the expression that Maple gives you (type the work infinity for an infinite sum).

Then press enter. Maple displays the appropriate summation expression.

For a decimal approximation, highlight the Maple output (the summation expression), right-click, and ask Maple for an approximation to an appropriate number of decimal positions.

Here is a sample, for the example in part 2 above:

Exercises for Part 3: Turn in a Maple worksheet with the following results:

1. Find S₁₀₀₀₀₀ for Get the results correct to 50 decimal positions

3. Use Maple to calculate the infinite sum , correct to 50 decimal places.

How many decimal places from S₁₀₀₀₀₀ were actually correct? ______________

Return to: Merced College; Don Power Updated 11/1/06 by Don Power