How do I find the radius of convergence of a series where n=1 to infinity of (14^(n)x^(n)n!)?
I have tried using the ratio test but I eventually get to this step: lim as n approaches infinity of absolute value of (14x(n+1)), which equal infinity. How am I to set the absolute value of infinity to less than 1 in order to find radius of convergence?
Any help would be appreciated!
Thanks in advance!
To find the radius of convergence of the series, we can use the ratio test. The ratio test states that for a power series of the form Σ(anxn), where an is a sequence of terms and x is the variable, the series converges absolutely if the following limit exists and is less than 1:
lim as n approaches infinity of |(an+1)/(an)| < 1.
In your case, the sequence of terms is given by an = 14^n * x^n * n!, so let's apply the ratio test to find the radius of convergence.
We have:
lim as n approaches infinity of |(an+1)/(an)| = lim as n approaches infinity of |(14^(n+1) * x^(n+1) * (n+1)!)/(14^n * x^n * n!)|.
We can simplify this expression by canceling out n! terms:
lim as n approaches infinity of |(14 * x * (n+1)) / n|.
Now we face the issue that the limit evaluates to infinity, and we want to compare it to 1 to determine the radius of convergence. In this case, it helps to consider the range of values that x can take.
Since x is a real number, it means that |14 * x * (n+1)| will eventually dominate the n term as n becomes very large. So, we can ignore the n term in the limit calculation and instead focus on the behavior of the expression |14 * x * (n+1)|.
Since the limit of |14 * x * (n+1)| as n approaches infinity is infinity for any nonzero value of x, we can conclude that the limit is always greater than 1.
Hence, for any nonzero value of x, the ratio |(an+1)/(an)| will always be greater than 1, indicating that the series diverges.
Therefore, the radius of convergence is 0, meaning that the series converges only when x = 0.
In summary, the series converges only when x = 0, and diverges for all other values of x.