Mister Exam

### Other calculators

• #### How to use it?

• Sum of series:
• 1/n
• sin(pi/n)
• x^(2n)/n!
• cos(n)/n
• #### Identical expressions

• x^(2n)/n!
• x to the power of (2n) divide by n!
• x(2n)/n!
• x2n/n!
• x^2n/n!
• x^(2n) divide by n!

# Sum of series x^(2n)/n!

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### The solution

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____
\
\     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
$$R^{2} = \infty$$
$$R = \infty$$
   /        / 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)`