Question

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{n(n+1)}$$

Short answer

The interval of convergence is $$\left(-1, 1\right]$$.

Step-by-step solution

Identify the Power Series

Given the power series: $$\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{n(n+1)}$$

Use the Ratio Test

To find the interval of convergence, use the Ratio Test, which involves the limit: $$\lim_{n \to \infty} \left|\frac{a_{n+1}(x)}{a_n(x)}\right|$$ where \(a_n = \frac{(-1)^{n} x^{n}}{n(n+1)}\).

Simplify the Ratio

Compute the ratio:$$\left|\frac{a_{n+1}(x)}{a_n(x)}\right| = \left|\frac{(-1)^{n+1} x^{n+1}}{(n+1)(n+2)} \cdot \frac{n(n+1)}{(-1)^{n} x^{n}}\right|$$ Simplifying gives: $$\left|x\right| \cdot \left|\frac{n}{n+2}\right|$$

Find the Limit

Take the limit as n approaches infinity: $$\lim_{n \to \infty} \left| x \cdot \frac{n}{n+2} \right| = \left| x \cdot \lim_{n \to \infty} \frac{n}{n+2} \right| = \left| x \cdot 1 \right| = \left| x \right|$$

Apply the Ratio Test Conclusion

For the series to converge, the above limit must be less than 1:$$\left| x \right| < 1$$Thus, the radius of convergence R = 1.

Check Endpoints

Investigate the behavior at the endpoints x = 1 and x = -1.For x = 1: $$\sum_{n=1}^{\infty} \frac{(-1)^{n} (1)^{n}}{n(n+1)} = -\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$$As this series diverges by the nth-term test.For x = -1:$$\sum_{n=1}^{\infty} \frac{(-1)^{n} (-1)^{n}}{n(n+1)} = \sum_{n=1}^{\infty} \frac{1}{n(n+1)}$$This series converges as it is a telescoping series.

Conclusion

Combining all results, the interval of convergence is:$$\left(-1, 1\right]$$

Key concepts

Power Series

A power series is a series of the form:
$$\textstyle \begin{equation*} \text { $$\textstyle \begin{equation*} \text {$$∑_{n=0}^{\text (infinity)}} 'a_n x^n.'$$ $$end{equation*}}$$ $$end{equation*}}$$


Here, 'a_n' are the coefficients, 'x' is the variable, and 'n' starts from 0 and goes to infinity.

Power series can represent a wide variety of functions. Understanding their convergence is crucial in many areas of mathematics, including calculus and analysis. When looking at power series, one important concept is their \html{interval of convergence}, \latex{\left[-1, 1\right)}.

. This interval is important because it tells us where the series will converge If the series converges at a particular value of 'x', the series sums up to a specific value. Otherwise, it diverges.

Ratio Test

The \textt{Ratio Test} is a popular method used to find the radius and interval of convergence for a power series. It involves checking the limit of the ratio of consecutive terms. Mathematically, it is expressed as:


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