The question is incomplete. To determine the radius of convergence, we need to know which function's Maclaurin series is being referred to. Let's explore the concept and provide examples with varying radii of convergence.
Understanding Radius of Convergence
The radius of convergence, often denoted as R, describes the interval in which a power series converges. Specifically, for a power series centered at a:
∑ cn(x - a)n
- If |x - a| < R, the series converges.
- If |x - a| > R, the series diverges.
- If |x - a| = R, the convergence needs to be checked individually (at the endpoints).
For Maclaurin series, the series is centered at a = 0.
Examples with Different Radii of Convergence
Here are some common Maclaurin series and their radii of convergence:
| Function | Maclaurin Series | Radius of Convergence (R) | Explanation |
|---|---|---|---|
| ex | ∑ (xn / n!) from n=0 to ∞ | ∞ | Converges for all real numbers. |
| sin(x) | ∑ ((-1)n x2n+1 / (2n+1)!) from n=0 to ∞ | ∞ | Converges for all real numbers. |
| cos(x) | ∑ ((-1)n x2n / (2n)!) from n=0 to ∞ | ∞ | Converges for all real numbers. |
| 1 / (1 - x) | ∑ xn from n=0 to ∞ | 1 | This is a geometric series that converges when |
| ln(1 + x) | ∑ ((-1)n+1 xn / n) from n=1 to ∞ | 1 | Can be found using the Ratio Test. |
| 1 / (1 + x2) | ∑ ((-1)n x2n) from n=0 to ∞ | 1 | Converges when |
| √(1+x) | 1 + (1/2)x - (1/8)x2 + (1/16)x3 - (5/128)x4 + ... | 1 | By Ratio Test and known expansion. |
| 1/(1-x2) | 1 + x2 + x4 + x6 +... = ∑ (x2)n from n=0 to ∞ | 1 | Converges for |
How to Find the Radius of Convergence
Common methods include:
- Ratio Test: Calculate limn→∞ |an+1 / an|. If the limit is L, the radius of convergence R = 1/L. If L = 0, R = ∞. If L = ∞, R = 0.
- Root Test: Calculate limn→∞ |an|1/n. If the limit is L, the radius of convergence R = 1/L.
In Summary
Without a specific function provided, we cannot determine the radius of convergence. The radius of convergence signifies the range of x-values for which the Maclaurin series converges. The examples illustrate that it can range from 0 to infinity.
