Question

Find the radius of convergence, R, of the series 1+2x+ 4!x ² / (2!)² + 6!x³ /(3!)² +8!x⁴/ (4!)² + 10!x⁵/(5!)²

Enter the xact value. R=

Answer 1

To find the radius of convergence, we can use the ratio test. By calculating the limit of the ratio of consecutive terms of the series, we can determine the range of x values for which the series converges. This range will give us the radius of convergence R.

Step-by-step explanation:

To find the radius of convergence of the series, we need to use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms of a series is less than 1, then the series converges. To apply the ratio test, let's calculate the ratio of consecutive terms:

r = |(2(n+1))!x^(n+1) / ((n+1)!)^2| / |(2n)!x^n / (n!)^2|

Next, simplify the expression and take the limit as n approaches infinity:

lim as n -> infinity of |(2(n+1))!x^(n+1) / ((n+1)!)^2| / |(2n)!x^n / (n!)^2| < 1

Simplifying further, we get:

lim as n -> infinity of (2n+2)(2n+1)/((n+1)^2)|x| < 1

Since we want the limit to be less than 1, we can ignore the absolute value. Solving the inequality, we get:

(2n+2)(2n+1)/(n+1)^2|x| < 1

Simplifying further, we get:

(4n^2 + 6n + 2)/(n^2 + 2n + 1) < 1

Now, we can solve for the range of x values for which the inequality holds true. This will give us the radius of convergence R.