Find the interval and radius of convergence for the

Question

1. Find the interval and radius of convergence for the following two power series

a. _n=1Σ^∞ 2ⁿxⁿ/n!

b.  _n=1Σ^∞ (x-3)ⁿ/2n+1

2. Find the third Taylor polynomial for f (x)= tanx centered at a = π/4.

3. a. Find the third Taylor polynomial for f (x) = sin x centered at π

b. What is the bound on the error if we use the Taylor polynomial from part a. to approximate sinx for

π- ¼ ≤ x ≤ π ≤ +1/4.

4. Use the Maclaurin series for tan^-1x to approximate ₀∫⁵ tan^-1x² dx with an error less than .001.

5. Give series representations (and their radius' of convergence) for the following functions, using known series:

a. x/3-x

b. 1/(3-x)²

c. ln(1 + x²)(start with 1/1+x)

6. Use the series you obtained in problem 5 c. to approximate ₀∫^1/2 ln(1+ x²)dx with an error less than .001.

7. Let f(x)= √(x+1)

a. Find T₂ , the second Taylor polynomial for f(x) centered at a = 0.

b. Get a bound for the error if we assume 0