3) Later in this course, we will learn that the function, arctan x, is equivalent to...

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3) Later in this course, we will learn that the function, arctan x, is equivalent to a power series for x on the interval 1≤x≤1: X -Σ Σ (-1)" 2n +1 n=0 arctan x= 2n+1 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? 3) Later in this course, we will learn that the function, arctan x, is equivalent to a power series for x on the interval 1≤x≤1: X -Σ Σ (-1)" 2n +1 n=0 arctan x= 2n+1 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?

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