We can use this power series to approximate the constant  . a) First, evaluate arctan(1) . (You do not need the series to evaluate it.) b) Use your answer from part (a) and the power series above to find a series representation for  . (The answer will be just 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  with |error| < 0.5. e) How many terms would you need to approximate  with |error| < 0.001?

(a)

(b)

(c)

Therefore if you sum any three terms of it you get the desired accuracy.

(d)

If you sum 1999 terms you get the desired accuracy.

Step-by-step explanation:

From the information given we know that

(a)

For that you need to find and angle such that = 1, remember that

therefore  

(b)

Then you just multiply by 4 and get that

(c)

Using the alternating series test, since the sequence    is decreasing and its limit tends to 0 when n tends to infinity the series is convergent.

(d)

Using the estimation theorem of alternating series we know that if denotes the partial sum of the series then

Therefore we are looking for an such that

we just have to solve that inequality, when you solve that inequality you get that

Therefore if you sum any three terms of it you get the desired accuracy.

(e) For this part you need to solve the following inequality

When you solve that inequality you get that

so, if you sum 1999 terms you get the desired accuracy.