Error in a popular AP Calc BC cheat sheet: arctan interval of convergence is wrong
I noticed an error in the popular Coach Forrester AP Calculus AB/BC cheat sheet:
https://coachforrester.weebly.com/uploads/1/3/1/9/13191763/ap_calculus_ab-bc_cheat_sheet.pdf
In the “BC Only: Common Series to MEMORIZE” section, it lists the Maclaurin series for arctan(x) as:
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
or generally:
arctan(x) = sum from n = 0 to infinity of (-1)^n * x^(2n+1)/(2n+1)
but the interval of convergence is shown as:
-infinity < x < infinity
That is incorrect.
The correct interval of convergence is:
[-1, 1]
The series comes from integrating the geometric series:
1/(1 + x^2) = 1 - x^2 + x^4 - x^6 + ...
which only converges when:
|x| < 1
After integrating, we get the arctan series. Then we check the endpoints.
At x = 1:
1 - 1/3 + 1/5 - 1/7 + ...
This converges by the alternating series test.
At x = -1:
-1 + 1/3 - 1/5 + 1/7 - ...
This also converges by the alternating series test.
So the correct interval is:
-1 <= x <= 1
or:
[-1, 1]
Important distinction: arctan(x) itself is defined for all real numbers, but its Maclaurin series only converges to arctan(x) on [-1, 1]. This could definitely confuse students studying for Calc BC, especially since tomorrow is the Calc BC test day.
Good luck to everyone taking the exam tomorrow!