Problem 4 - Calculus Early Transcendentals

Question

How do you find the interval of convergence of a Taylor series?

Step-by-Step Solution

Answer

Question: Determine the interval of convergence for the given Taylor series. Answer: To find the interval of convergence, apply the Ratio Test and follow these steps: 1. Use the limit as n approaches infinity of the absolute value of the ratio of consecutive terms in the Taylor series. 2. Simplify the ratio inside the absolute value. 3. Determine the condition for convergence of the series by setting the limit to be less than 1. 4. Solve for the radius of convergence (R). 5. Find the interval of convergence, and check the endpoints by plugging in x = a-R and x = a+R into the Taylor series to see if it converges. 6. Write the final answer as the interval of convergence in parentheses or brackets.

Step 1: Apply the Ratio Test

To begin, apply the Ratio Test to the given Taylor series. Take the limit as n approaches infinity of the absolute value of the ratio of consecutive terms: \(\lim_{n\to\infty} \left| \frac{a_{n+1}(x-a)^{n+1}}{a_n(x-a)^n} \right|\). 2.

Step 2: Simplify the Ratio

Simplify the ratio inside the absolute value: $$\lim_{n\to\infty} \left|\frac{a_{n+1}(x-a)^{n+1}}{a_n(x-a)^n} \right| = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|\cdot|(x-a)|$$ 3.

Step 3: Determine the Convergence Condition

According to the Ratio Test, if the limit is less than 1, then the series converges. Therefore, we need to find when: $$\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|\cdot|(x-a)| < 1$$ 4.

Step 4: Solve for the Radius of Convergence

Solve for the radius of convergence (denoted by \(R\)): $$R = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right|$$ 5.

Step 5: Find the Interval of Convergence

After finding the radius of convergence R, we can find the interval of convergence as \((a-R, a+R)\). Check the endpoints by plugging \(x = a-R\) and \(x = a+R\) into the Taylor series and determining if it converges at those points. 6.

Step 6: Write the Final Answer

Using the information found in steps 4 and 5, write the interval of convergence in either parentheses or brackets, parentheses indicating that the endpoint is not included, and brackets indicating it is included. For example, the interval of convergence could look like \((a-R, a+R)\), \([a-R, a+R)\), or \((a-R, a+R]\).