# How do you find the radius of convergence of the power series #Sigma (n!)/(n^n)x^(2n)# from #n=[1,oo)#?

The radius of convergence is

We can apply the ratio test, stating that a series:

is absolutely convergent if:

Calculate the expression of the ratio for this series:

Now we have:

so that:

The series is then absolutely convergent if:

By signing up, you agree to our Terms of Service and Privacy Policy

To find the radius of convergence of the power series (\sum_{n=1}^{\infty} \frac{n!}{n^n} x^{2n}), we use the ratio test.

The ratio test states that if (L) is the limit of (|\frac{a_{n+1}}{a_n}|) as (n) approaches infinity, then:

- If (L < 1), the series converges absolutely.
- If (L > 1), the series diverges.
- If (L = 1), the test is inconclusive.

In this case, let (a_n = \frac{n!}{n^n} x^{2n}). Then:

[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} \cdot \frac{x^{2(n+1)}}{x^{2n}} \right|]

[= \lim_{n \to \infty} \left| \frac{(n+1)n!}{(n+1)^{n+1}} \cdot \frac{x^2}{1} \right|]

[= \lim_{n \to \infty} \left| \frac{n!}{(n+1)^{n}} \cdot \frac{x^2}{1} \right|]

[= \lim_{n \to \infty} \left| \frac{1}{\left(1+\frac{1}{n}\right)^{n}} \cdot \frac{x^2}{1} \right|]

[= \frac{x^2}{e}]

To ensure convergence, we require (L < 1):

[\frac{|x|^2}{e} < 1]

[|x|^2 < e]

[|x| < \sqrt{e}]

Therefore, the radius of convergence is (\sqrt{e}).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you use the limit comparison test for #sum ((sqrt(n+1)) / (n^2 + 1))# as n goes to infinity?
- How do you apply the ratio test to determine if #Sigma (4^n(n!)^2)/((2n)!)# from #n=[1,oo)# is convergent to divergent?
- Find the limit of the sequence an=2^n/(2n-1)?
- How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (-1)^(n+1)/(n+1)^2# from #[1,oo)#?
- How do you show whether the improper integral #int lim (lnx) / x dx# converges or diverges from 1 to infinity?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7