To find the radius of convergence of a power series, $\sum_{n=0}^{\infty} c_n x^n$, the ratio and root tests are commonly used. Here’s how each test applies:

1. Using the Ratio Test

For the ratio test, we examine the limit:

$L = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right|$

• Interpretation for Radius of Convergence $R$:

  • If $L = 0$, the series converges for all $x$, so $R = \infty$.

  • If $L = \infty$, the series only converges at $x = 0$, so $R = 0$.

  • If $0 < L < \infty$, the radius of convergence $R$ is given by:

    $R = \frac{1}{L}$

2. Using the Root Test

The root test examines the limit:

$L = \lim_{n \to \infty} \sqrt[n]{|c_n|}$

• Interpretation for Radius of Convergence $R$:

  • If $L = 0$, the series converges for all $x$, so $R = \infty$.

  • If $L = \infty$, the series only converges at $x = 0$, so $R = 0$.

  • If $0 < L < \infty$, then the radius of convergence $R$ is given by:

    $R = \frac{1}{L}$

Summary:

Both tests provide a way to calculate the radius of convergence, but the choice depends on the structure of the series terms $c_n$. The ratio test is often simpler when dealing with factorials or linear growth terms, while the root test can be more effective when terms grow exponentially.

Would you like a worked example to see these in action, or any specific part explained further?

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Related Questions:

1. How does the radius of convergence affect the interval of convergence?
2. Can the ratio and root tests give different results for the same series?
3. How does convergence at endpoints affect the interval of convergence?
4. Are there specific series types better suited for the ratio test over the root test?
5. Can the radius of convergence be zero? Under what conditions?

Tip:

For power series with factorials in $c_n$, the ratio test is typically simpler, while the root test works well with exponential terms