Mister Exam
Other calculators
Sum of series x^(2n)/n!
The solution
You have entered
[src]
oo ____ \ ` \ 2*n \ x / ---- / n! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{x^{2 n}}{n!}$$
Sum(x^(2*n)/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{x^{2 n}}{n!}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 2$$
,
$$c = 1$$
then
$$R^{2} = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|$$
Let's take the limit
we find
$$R^{2} = \infty$$
$$R = \infty$$
$$\frac{x^{2 n}}{n!}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 2$$
,
$$c = 1$$
then
$$R^{2} = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|$$
Let's take the limit
we find
$$R^{2} = \infty$$
$$R = \infty$$
The answer
[src]
/ / 2\\ | \x /| 2 | 1 e | x *|- -- + -----| | 2 2 | \ x x /
$$x^{2} \left(\frac{e^{x^{2}}}{x^{2}} - \frac{1}{x^{2}}\right)$$
x^2*(-1/x^2 + exp(x^2)/x^2)