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  • Sum of series:
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  • (-1)^nx^(2n)/(2n)!
  • sinn/sqrtn sinn/sqrtn
  • 5/3^n 5/3^n

  • Identical expressions

  • (- one)^nx^(2n)/(2n)!
  • ( minus 1) to the power of nx to the power of (2n) divide by (2n)!
  • ( minus one) to the power of nx to the power of (2n) divide by (2n)!
  • (-1)nx(2n)/(2n)!
  • -1nx2n/2n!
  • -1^nx^2n/2n!
  • (-1)^nx^(2n) divide by (2n)!

  • Similar expressions

  • (1)^nx^(2n)/(2n)!
Sum of series/ (-1)^nx^(2n)/(2n)!

Sum of series (-1)^nx^(2n)/(2n)!


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \        n  2*n
  \   (-1) *x   
  /   ----------
 /      (2*n)!  
/___,           
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n} x^{2 n}}{\left(2 n\right)!}$$ 
Sum(((-1)^n*x^(2*n))/factorial(2*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n} x^{2 n}}{\left(2 n\right)!}$$
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
$$a_{n} = \frac{\left(-1\right)^{n}}{\left(2 n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 2$$
,
$$c = 1$$
then
$$R^{2} = \lim_{n \to \infty} \left|{\frac{\left(2 n + 2\right)!}{\left(2 n\right)!}}\right|$$
Let's take the limit
we find
$$R^{2} = \infty$$
$$R = \infty$$
The answer [src] 
  2 /2    2*cos(x)\ 
-x *|-- - --------| 
    | 2       2   | 
    \x       x    / 
--------------------
         2
$$- \frac{x^{2} \left(- \frac{2 \cos{\left(x \right)}}{x^{2}} + \frac{2}{x^{2}}\right)}{2}$$ 
-x^2*(2/x^2 - 2*cos(x)/x^2)/2

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