## Problems

Answers to the following problems are found in the folder "Ch. 04 (Number Series)" in the "Problems & Solutions" folder on the CD.

1. Evaluate the following infinite series:

2. Evaluate the following:

3. Evaluate the following infinite series:

4. Evaluate the following:

5. Evaluate the following:

6. Evaluate Wallis' series for n:

{in - lX2« +1) over the first 100 terms of the series.

7. Evaluate Wallis' series for %, summing over 65,536 terms. Use a worksheet formula that uses ROW and INDIRECT to create the series of integers.

8. A simple yet surprisingly efficient method to calculate the square root of a number is variously called Heron's method, Newton's method, or the divide-and-average method. To find the square root of the number a:

1. Begin with an initial estimate x.

2. Divide the number by the estimate (i.e., evaluate a/x), to get a new estimate

3. Average the original estimate and the new estimate (i.e., (x + a/x)/2) to get a new estimate

4. Return to step 2. Use this method to calculate the square root of a number. The value of the initial estimate x must be greater than zero.

9. In the divide-and-average method, the better the initial estimate, the faster the convergence. Devise an Excel formula to provide an effective initial estimate.

10. The series proposed by Machin in 1706, converges quickly. Determine the value of it to 15 digits by using this series

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