- #1
thomas12323
- 2
- 0
Homework Statement
The problem is looking for the radius of convergence
sum(5^n)((x-3)^n)/n
n=1
The formula for the Ratio Test for Radius of Convergence is given by:
R = lim_{n→∞} |a_{n}|/|a_{n+1}|, where a_{n} is the nth term of the series.
To apply the Ratio Test for Radius of Convergence, first find the nth term of the series. Then, take the absolute value of the nth term and divide it by the absolute value of the (n+1)th term. Finally, take the limit of this ratio as n approaches infinity. If the limit is less than 1, the series is absolutely convergent, and if the limit is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive and another test must be used.
No, the Ratio Test for Radius of Convergence can only be used on series with positive terms. It cannot be used on alternating series or series with negative terms.
To find the radius of convergence using the Ratio Test, first apply the test to the series. If the limit is less than 1, the radius of convergence is infinite. If the limit is greater than 1, the radius of convergence is 0. If the limit is equal to 1, the radius of convergence can be found by using the formula: R = 1/lim_{n→∞} |a_{n}|^(1/n).
No, the Ratio Test for Radius of Convergence can only determine if a series is absolutely convergent or divergent. To determine if a series is conditionally convergent, the Alternating Series Test or the Ratio Test for Alternating Series must be used.