Mathematics 210 - Spring term 2005 - Fourth assignment

This assignment requires you to submit a few more spreadsheets concerned with series. It is due before class time Monday, February 7. There is some calculation to do by hand, which should be handed in on the Commnets window.

The basic point of the first three questiosn is to compute the Taylor series of arc sin(x) at x=0. This is hard to do directly, but arc sin is the integral of 1/sqrt(1 - x2), and a power series for this can be found from the Taylor series of 1/sqrt(x) at 1.

• Question 1. Let f(x) = 1/sqrt(x)). Lay out in column a the numbers n = 0, 1, 2, etc. and in column b the first 25 derivatives of f(x) at x=1. Do this by finding a recursion formula for f(n+1) in terms of n and f(n). Then lay out in column c the first 25 coefficients in the Taylor's series of f(x) at 1, also using a recursion formula.

Save as m210.4.1.ms. Save this work for the next question.

• Question 2. Use the columns from the previous exercise to lay out in column d the terms in the Taylor series giving f(1-x2) around x=0, where x is stored in m0. Use the integral formula to give in column e the Taylor series for arc sin(x), and in column f the partial sums of this series. Set x = 1/2. Tell us in the comment box what the exact value of arc sin(1/2) is.

Save the spread sheet as m210.4.2.ms. Save this work for the next question.

• Question 3. Use the columns from the previous exercise. Lay out enough rows to calculate the value of arc sin(x) to an acccuracy of 10 decimals. Explain how you decided this in the comments box.

Then compute the same sum in the next free column, but by accumulating terms from the smallest up.

Save the spread sheet as m210.4.3.ms.

• Question 4. Lay out a spreadsheet to calculate the sum of terms 1/n8. Estimate how many terms are needed to calculate the sum to 16 decimals, and lay out enough rows to do that. Sum both forwards and backwards. Comment: how many terms would be required if you could and wanted to do 32 decimals.

Save the spread sheet as m210.4.4.ms.

If you find these questions confusing, please write me.