Question

Use this list of Basic Taylor Series to find the Taylor Series for tan-1(x) based at 0. Give your answer using summation notation, write out the first three non-zero terms, and give the interval on which the series converges. (if you need to enter 00, use the 00 button in CalcPad or type "infinity" in all lower-case.) The Taylor series for tan -1(x) is: The first three non-zero terms are: + + + The Taylor series converges to tan-1(x) for |x| < . Use the Taylor series you just found for tan -1(x)n to find the Taylor series for f(x) = tan -1(t) dt based at 0. Give your answer using summation notation, write out the first three non-zero terms, and given the interval on which the series converges. The Taylor series for f(x) = tan -1(t) dt is: The first series converges to f(x) for |x| < .

Answer 1

First we need to use the following series:

which converges for |x| < 1.

Now, remember that:

which means that :

k=0

Integration nor differentiation changes the radius of convergence so the series will converge for

|x| < 1.

k=0

3 x 4 5X6 6X7

Again, integration won't change where the series converges so it will converge for |x| < 1.