Question

We can use this power series to approximate the constant pi:

arctan(x) = (summation from n = 1 to infinity) of ((-1)^n * x^(2n+1))/(2n+1)

a) First evaluate arctan(1) without the given series. (I know this is pi/4)
b) Use your answer from part (a) and the power series to find a series representation for pi (a series, not a power series).
c) Verify that the series you found in part (b) converges.
d) Use your convergent series from part (b) to approximate pi with |error| < 0.5.
e) How many terms would you need to approximate pi with |error| < 0.001?

You don't have to provide answers with c, d, and e. Just some logic/methods to help me solve them would be great. It's just part (b) that I'm having a hard time to set up.

Thanks in advance!

Answer 1

since pi/4 = arctan(1) just plug in x=1 into the power series. That gives the actual series.

Answer 2

Thanks, Steve. Now I got parts b and c down.

How would I go about doing parts d and e?