Question

Taylor series and interval of convergencec. Determine the interval of convergence of the series.f(x) = ln (x − 2), a = 3

Accepted Answer

Start by writing the Taylor series expansion of the function $$f(x) = \ln(x - 2)$$ centered at $$a = 3. $$Recall that the Taylor series of $$f(x)$$ about $$a is $$given by $$\displaystyle \sum_{n=0}^{\infty} \frac{f$$^{(n)}$$(a)}{n!} (x - a)^n. $$Find the derivatives of $$f(x) = \ln(x - 2)$$ evaluated at $$x = 3. $$Note that $$f(x)$$ can be rewritten as $$\ln(u)$$ where $$u = x - 2$$, so $$f(x) = \ln(u)$$ and $$u is $$shifted by 1 at $$x=3$$ since $$u = 3 - 2 = 1. $$Express the general term of the Taylor series using the $$n-th $$derivative of $$f at$$ $$a=3. $$For $$f(x) = \ln(x-2)$$, the $$n-th $$derivative has a known pattern involving factorials and powers of $$(x-2)$$, which you can use to write the general term $$c_n (x-3)^n. $$Determine the radius of convergence $$R of $$the Taylor series by applying the Ratio Test or Root Test to the general term of the series. This involves taking the limit as $$n 	o \infty of $$the ratio of consecutive terms or the $$n-th $$root of the absolute value of the $$n-th $$term. Once you find the radius of convergence $$R$$, write the interval of convergence as $$(a - R, a + R)$$, i.e., $$(3 - R, 3 + R). $$Then check the endpoints $$x = 3 - R$$ and $$x = 3 + R by $$substituting back into the original series to see if the series converges or diverges at these points, thus determining the full interval of convergence.

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = ln (x − 2), a = 3

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Key Concepts

Taylor Series Expansion

Radius and Interval of Convergence

Domain Restrictions and Singularities