In the 16th and 17th centuries the calculation of π was changed dramatically with the introduction of infinite series. For example, the following series is called the Gregory-Leibniz series and is equal to π.
π = (4/1) - (4/3) + (4/5)-(4/7) + (4/9)-…..
In theory, this allows a mathematician to produce arbitrarily good approximations to π using only the four basic numerical operations by computing very large partial sums. The reality of the situation is a bit more complicated. Answer the following questions in paragraph form. Be sure to show your work.

Question

In the 16th and 17th centuries the calculation of π was changed dramatically with the introduction of infinite series. For example, the following series is called the Gregory-Leibniz series and is equal to π.
π = (4/1) - (4/3) + (4/5)-(4/7) + (4/9)-…..
In theory, this allows a mathematician to produce arbitrarily good approximations to π using only the four basic numerical operations by computing very large partial sums. The reality of the situation is a bit more complicated. Answer the following questions in paragraph form. Be sure to show your work.

anonymous · Answer

1) Write down the series using ∑-notation?

2) Does this series converge absolutely or converge conditionally? Show your work for any of the tests you use to demonstrate convergence.

3) Use a partial sum with 10 terms to approximate the value of the series. How many digits of π did you get correct?

4) How many terms do you need if you want a partial sum which approximates π to at least 100 decimal places (i.e. with an error of 10^100)?

5) The worlds fastest computer (currently the Tianhe-2 of China) could compute partial sums for this series at a rate of about 6:66 * 10^17 seconds per term. How many years would it take to compute an approximation to π which is correct to 100 decimal places using such a computer.