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How Do You Use the Ratio Test to Find the Radius of Convergence?

To find the radius of convergence for power series, we can use a helpful method called the Ratio Test. This test helps us figure out the values of (x) where the series works.

A power series usually looks like this:

[ \sum_{n=0}^{\infty} a_n (x - c)^n, ]

In this formula, (a_n) are numbers we use, (x) is our variable, and (c) is the center of convergence. Our goal is to find the interval around (c) where the series converges.

Steps to Use the Ratio Test:

  1. Identify the Terms: Start with the general term of the power series, which we write as (a_n (x - c)^n). To use the Ratio Test, we need to look at the absolute value of the ratio of the terms:

[ \left| \frac{a_{n+1} (x - c)^{n+1}}{a_n (x - c)^n} \right|. ]

  1. Simplify the Ratio: We can simplify this ratio to:

[ \left| \frac{a_{n+1}}{a_n} \right| \cdot |x - c|. ]

  1. Find the Limit: Next, we find the limit of this ratio as (n) gets really big:

[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|. ]

  1. Use the Ratio Test: According to the Ratio Test, the series converges if:

[ L |x - c| < 1. ]

We can rearrange this to find the radius of convergence, (R):

[ |x - c| < \frac{1}{L}. ]

So, the radius of convergence (R) is:

[ R = \frac{1}{L}. ]

  1. Convergence and Divergence: The series will not work (diverge) if:

[ L |x - c| > 1, ]

and it may converge sometimes if (L |x - c| = 1).

Quick Summary of Steps:

  • Identify the power series and the coefficients (a_n).
  • Calculate the ratio (\left| \frac{a_{n+1}}{a_n} \right|).
  • Determine the limit (L) as (n) gets big.
  • Find the radius of convergence with (R = \frac{1}{L}).
  • Check whether it converges or diverges based on (L |x - c|).

Special Cases:

  • If (L = 0) (when the coefficients (a_n) go to 0), the series converges for all (x) (so the radius is infinite).
  • If (L = \infty), the series converges only at the center (c) (meaning the radius is zero).

Interval of Convergence:

Once we find the radius (R), we can express the interval of convergence like this:

[ (c - R, c + R). ]

It's important to check the endpoints (x = c - R) and (x = c + R) separately because the Ratio Test doesn't work at these points.

When checking these endpoints, we often use other tests like the Alternating Series Test or the Direct Comparison Test, depending on the series.

In conclusion, the Ratio Test gives a clear way to see if power series converge by calculating a limit and then using that to find the radius of convergence. Understanding this method is important for exploring power series in calculus.

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How Do You Use the Ratio Test to Find the Radius of Convergence?

To find the radius of convergence for power series, we can use a helpful method called the Ratio Test. This test helps us figure out the values of (x) where the series works.

A power series usually looks like this:

[ \sum_{n=0}^{\infty} a_n (x - c)^n, ]

In this formula, (a_n) are numbers we use, (x) is our variable, and (c) is the center of convergence. Our goal is to find the interval around (c) where the series converges.

Steps to Use the Ratio Test:

  1. Identify the Terms: Start with the general term of the power series, which we write as (a_n (x - c)^n). To use the Ratio Test, we need to look at the absolute value of the ratio of the terms:

[ \left| \frac{a_{n+1} (x - c)^{n+1}}{a_n (x - c)^n} \right|. ]

  1. Simplify the Ratio: We can simplify this ratio to:

[ \left| \frac{a_{n+1}}{a_n} \right| \cdot |x - c|. ]

  1. Find the Limit: Next, we find the limit of this ratio as (n) gets really big:

[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|. ]

  1. Use the Ratio Test: According to the Ratio Test, the series converges if:

[ L |x - c| < 1. ]

We can rearrange this to find the radius of convergence, (R):

[ |x - c| < \frac{1}{L}. ]

So, the radius of convergence (R) is:

[ R = \frac{1}{L}. ]

  1. Convergence and Divergence: The series will not work (diverge) if:

[ L |x - c| > 1, ]

and it may converge sometimes if (L |x - c| = 1).

Quick Summary of Steps:

  • Identify the power series and the coefficients (a_n).
  • Calculate the ratio (\left| \frac{a_{n+1}}{a_n} \right|).
  • Determine the limit (L) as (n) gets big.
  • Find the radius of convergence with (R = \frac{1}{L}).
  • Check whether it converges or diverges based on (L |x - c|).

Special Cases:

  • If (L = 0) (when the coefficients (a_n) go to 0), the series converges for all (x) (so the radius is infinite).
  • If (L = \infty), the series converges only at the center (c) (meaning the radius is zero).

Interval of Convergence:

Once we find the radius (R), we can express the interval of convergence like this:

[ (c - R, c + R). ]

It's important to check the endpoints (x = c - R) and (x = c + R) separately because the Ratio Test doesn't work at these points.

When checking these endpoints, we often use other tests like the Alternating Series Test or the Direct Comparison Test, depending on the series.

In conclusion, the Ratio Test gives a clear way to see if power series converge by calculating a limit and then using that to find the radius of convergence. Understanding this method is important for exploring power series in calculus.

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