Determine the Interval of Convergence for the Series

Oct 13, 2025 by miodeya.com 54 views

Determine the Interval of Convergence for the Series

The series $\sum_{n=1}^{\infty} \frac{(x-1)^n}{n^2}$ is a power series, which is a series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$ . In this case, the series is centered at $x=1$ and has coefficients $a_n = \frac{1}{n^2}$ . The interval of convergence is the set of all values of $x$ for which the series converges.

To determine the interval of convergence, we can use the ratio test. The ratio test states that if $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$ , then the series converges. In this case, we have

\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty} \left|\frac{\frac{1}{(n+1)^2}}{\frac{1}{n^2}}\right| = \lim_{n\to\infty} \left|\frac{n^2}{(n+1)^2}\right| = \lim_{n\to\infty} \left|\frac{n^2}{n^2 + 2n + 1}\right| = 1.

Since the limit is equal to 1, the ratio test is inconclusive, and we need to use other methods to determine the interval of convergence.

Using the Root Test

Another method to determine the interval of convergence is to use the root test. The root test states that if $\lim_{n\to\infty} \sqrt[n]{|a_n|} < 1$ , then the series converges. In this case, we have

\lim_{n\to\infty} \sqrt[n]{|a_n|} = \lim_{n\to\infty} \sqrt[n]{\frac{1}{n^2}} = \lim_{n\to\infty} \frac{1}{n^{2/n}} = 1.

Since the limit is equal to 1, the root test is also inconclusive.

Using the Comparison Test

We can use the comparison test to determine the interval of convergence. The comparison test states that if $\sum a_n$ and $\sum b_n$ are two series such that $|a_n| \leq |b_n|$ for all $n$ , and $\sum b_n$ converges, then $\sum a_n$ also converges. In this case, we can compare the series $\sum_{n=1}^{\infty} \frac{(x-1)^n}{n^2}$ with the series $\sum_{n=1}^{\infty} \frac{|x-1|^n}{n^2}$ .

Since $|x-1|^n \geq (x-1)^n$ for all $x$ , we have

\left|\frac{(x-1)^n}{n^2}\right| \leq \frac{|x-1|^n}{n^2}.

The series $\sum_{n=1}^{\infty} \frac{|x-1|^n}{n^2}$ converges by the ratio test, since

\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty} \left|\frac{\frac{|x-1|^{n+1}}{(n+1)^2}}{\frac{|x-1|^n}{n^2}}\right| = \lim_{n\to\infty} \left|\frac{|x-1|n^2}{(n+1)^2}\right| = \lim_{n\to\infty} \left|\frac{|x-1|}{\left(1+\frac{1}{n}\right)^2}\right| = |x-1| < 1.

Therefore, the series $\sum_{n=1}^{\infty} \frac{(x-1)^n}{n^2}$ also converges.

Determining the Interval of Convergence

We have shown that the series $\sum_{n=1}^{\infty} \frac{(x-1)^n}{n^2}$ converges for all $x$ such that $|x-1| < 1$ . This means that the series converges for all $x$ in the interval $(-1, 1)$ .

However, we need to check the endpoints of the interval, $x=-1$ and $x=1$ . When $x=-1$ , the series becomes $\sum_{n=1}^{\infty} \frac{(-2)^n}{n^2}$ , which diverges by the ratio test, since

\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty} \left|\frac{\frac{(-2)^{n+1}}{(n+1)^2}}{\frac{(-2)^n}{n^2}}\right| = \lim_{n\to\infty} \left|\frac{(-2)n^2}{(n+1)^2}\right| = \lim_{n\to\infty} \left|\frac{-2}{\left(1+\frac{1}{n}\right)^2}\right| = 2 > 1.

When $x=1$ , the series becomes $\sum_{n=1}^{\infty} \frac{0}{n^2}$ , which converges to 0.

Therefore, the interval of convergence is $(-1, 1]$ .

Conclusion

In conclusion, the series $\sum_{n=1}^{\infty} \frac{(x-1)^n}{n^2}$ converges for all $x$ in the interval $(-1, 1]$ . This means that the correct answer is C. $[-1,1]$ .
Q&A: Determining the Interval of Convergence for the Series

In the previous article, we determined the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(x-1)^n}{n^2}$ . In this article, we will answer some common questions related to determining the interval of convergence for a power series.

Q: What is the interval of convergence for a power series?

A: The interval of convergence for a power series is the set of all values of $x$ for which the series converges. In other words, it is the set of all values of $x$ for which the series $\sum_{n=0}^{\infty} a_n(x-c)^n$ converges.

Q: How do I determine the interval of convergence for a power series?

A: There are several methods to determine the interval of convergence for a power series, including:

Using the ratio test
Using the root test
Using the comparison test
Using the limit comparison test

Q: What is the ratio test?

A: The ratio test is a method to determine the interval of convergence for a power series. It states that if $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$ , then the series converges. If the limit is greater than 1, then the series diverges.

Q: What is the root test?

A: The root test is a method to determine the interval of convergence for a power series. It states that if $\lim_{n\to\infty} \sqrt[n]{|a_n|} < 1$ , then the series converges. If the limit is greater than 1, then the series diverges.

Q: What is the comparison test?

A: The comparison test is a method to determine the interval of convergence for a power series. It states that if $\sum a_n$ and $\sum b_n$ are two series such that $|a_n| \leq |b_n|$ for all $n$ , and $\sum b_n$ converges, then $\sum a_n$ also converges.

Q: What is the limit comparison test?

A: The limit comparison test is a method to determine the interval of convergence for a power series. It states that if $\sum a_n$ and $\sum b_n$ are two series such that $\lim_{n\to\infty} \frac{a_n}{b_n} = c$ , where $c$ is a positive constant, then either both series converge or both series diverge.

Q: How do I determine the interval of convergence for a power series using the ratio test?

A: To determine the interval of convergence for a power series using the ratio test, you need to calculate the limit of the ratio of consecutive terms. If the limit is less than 1, then the series converges. If the limit is greater than 1, then the series diverges.

Q: How do I determine the interval of convergence for a power series using the root test?

A: To determine the interval of convergence for a power series using the root test, you need to calculate the limit of the nth root of the absolute value of the nth term. If the limit is less than 1, then the series converges. If the limit is greater than 1, then the series diverges.

Q: How do I determine the interval of convergence for a power series using the comparison test?

A: To determine the interval of convergence for a power series using the comparison test, you need to find a series that is known to converge or diverge, and compare it to the power series. If the power series is less than or equal to the known series, then the power series converges.

Q: What are some common mistakes to avoid when determining the interval of convergence for a power series?

A: Some common mistakes to avoid when determining the interval of convergence for a power series include:

Not checking the endpoints of the interval
Not using the correct test (e.g. using the ratio test when the root test is more appropriate)
Not calculating the limit correctly
Not considering the case where the series converges or diverges at the endpoints of the interval

Conclusion

In conclusion, determining the interval of convergence for a power series is an important step in understanding the behavior of the series. By using the ratio test, root test, comparison test, and limit comparison test, you can determine the interval of convergence for a power series. Remember to check the endpoints of the interval and to avoid common mistakes when determining the interval of convergence.