Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. If p(x) is the Taylor series for f centered at 0, then p(x−1) is the Taylor series for f centered at 1.

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Recall that the Taylor series for a function \(f\) centered at a point \(a\) is given by the formula: \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\] where \(f^{(n)}(a)\) denotes the \(n\)-th derivative of \(f\) evaluated at \(a\).

Given that \(p(x)\) is the Taylor series for \(f\) centered at 0, it means \[p(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n.\]

Now, consider \(p(x-1)\). Substituting \(x-1\) into the series, we get \[p(x-1) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (x-1)^n.\]

To check if \(p(x-1)\) is the Taylor series for \(f\) centered at 1, compare it with the Taylor series centered at 1: \[\sum_{n=0}^{\infty} \frac{f^{(n)}(1)}{n!} (x-1)^n.\] Notice that the coefficients involve derivatives evaluated at 1, not at 0.

Therefore, \(p(x-1)\) uses derivatives at 0, not at 1, so it generally does not equal the Taylor series of \(f\) centered at 1. This shows that the statement is false, and \(p(x-1)\) is not the Taylor series for \(f\) centered at 1.

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Taylor Series and Center of Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a specific point called the center. The series approximates the function near this center, and changing the center changes the coefficients and form of the series.

Effect of Function Translation on Taylor Series

Replacing x by (x - a) in a Taylor series centered at 0 shifts the input but does not automatically produce the Taylor series centered at a. The coefficients depend on derivatives at the new center, so simply substituting variables does not yield the correct series centered at a.

Counterexamples in Series Expansion

To test if p(x−1) is the Taylor series of f centered at 1, one can use specific functions like e^x or sin x. These examples show that substituting x−1 into p(x) does not match the Taylor series centered at 1, illustrating the statement is false.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.d. If p(x) is the Taylor series for f centered at 0, then p(x−1) is the Taylor series for f centered at 1.