Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.




x +x³/3 +x⁵/5 +x⁷/7 + ...

Verified step by step guidance

1. Identify the general term of the power series. Notice the pattern in the series: the powers of \( x \) are odd numbers (1, 3, 5, 7, ...) and the denominators are the same odd numbers. So the general term can be written as \( a_n = \frac{x^{2n+1}}{2n+1} \) where \( n = 0, 1, 2, \ldots \).
2. Apply the Ratio Test to the general term to find the radius of convergence. The Ratio Test involves computing the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). Substitute the general term expressions for \( a_{n+1} \) and \( a_n \) and simplify the expression.
3. Simplify the ratio inside the limit: \( \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{x^{2(n+1)+1}}{2(n+1)+1}}{\frac{x^{2n+1}}{2n+1}} \right| = \left| x^2 \right| \cdot \frac{2n+1}{2n+3} \). Then take the limit as \( n \to \infty \) to find \( L = |x|^2 \cdot 1 = |x|^2 \).
4. Use the Ratio Test criterion for convergence: the series converges if \( L < 1 \), which means \( |x|^2 < 1 \) or equivalently \( |x| < 1 \). This gives the radius of convergence \( R = 1 \).
5. Test the endpoints \( x = -1 \) and \( x = 1 \) by substituting them into the original series and checking for convergence. This typically involves recognizing the resulting series (e.g., alternating series or p-series) and applying appropriate convergence tests such as the Alternating Series Test or the p-series test.

Key Concepts

Radius of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges absolutely. It defines the interval on the real line where the series sums to a finite value. Finding this radius helps determine where the power series represents a valid function.

Ratio and Root Tests

The Ratio Test and Root Test are methods to determine the convergence of infinite series. The Ratio Test examines the limit of the ratio of successive terms, while the Root Test looks at the nth root of the absolute value of terms. Both tests help find the radius of convergence for power series.

Testing Endpoints for Interval of Convergence

After finding the radius of convergence, the interval of convergence includes points within that radius. However, convergence at the endpoints must be checked separately, as the series may converge or diverge there. Testing endpoints ensures the full interval of convergence is accurately identified.