Use the substitution t = x - a to find the Taylor series at the indicated value of a. State the interval of convergence for the
series.
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Question

Quizplus · Accepted Answer

Starting with the substitution $t = x - a$, we have $x = t + a$. Substituting this into the given function $f(x) = \ln(x)$, we get $f(t+a) = \ln(t+a)$. Now we can use the Taylor series for $\ln(1+x)$ centered at $x=0$, which is:

$$\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} x^n, \quad -1 < x \leq 1$$

Using this with $x=t+a$, we get:

$$f(t+a) = \ln(t+a) = \ln(1+(t+a-1)) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} (t+a-1)^n, \quad -1 < t+a-1 \leq 1$$

Simplifying the inequality, we get $-2 < t+a < 0$, which means the interval of convergence is $\boxed{	ext{(B) } -2 < t < 0}$.

Use the Substitution T = X - a to Find