Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Let pn be the nth-order Taylor polynomial for f centered at 2. The approximation p3 (2.1) ≈ f(2.I ) is likely to be more accurate than the approximation p2 (2.2) ≈ f(2.2).
b. If the Taylor series for f centered at 3 has a radius of convergence of 6. then the interval of convergence is [−3. 9].
c. The interval of convergence of the power serie; Σck xk could
d. The Taylor series for f(x)= (1 + .r)12 centered at 0 has a finite number of terms.
In this problem we have to explain whether the given statements are true or false.Given statement is “ is the order Taylor polynomial for centered at 2 then the approximation is more accurate than ”
The given statement is true .Because 2.1 is nearer to 2 than 2.2
b) Given statement is “ the Taylor series for f centered at 3 has a radius of convergence of 6 , then the interval of convergence is [-3,9]”.
The given statement is false.
Because according to given data |x-3| < 6.
That is , -6 < (x - 3) < 6
Therefore , the interval of convergence is (-3,9).