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  • Sum of series:
  • 1/4+1/28+1/70 1/4+1/28+1/70
  • ((-1)^n)*x^2n/(2n+1)!
  • ln(k+1)/n
  • -6n(6n+6)+(6n-6)(6+6n) -6n(6n+6)+(6n-6)(6+6n)

  • Identical expressions

  • ((- one)^n)*x^2n/(2n+ one)!
  • (( minus 1) to the power of n) multiply by x squared n divide by (2n plus 1)!
  • (( minus one) to the power of n) multiply by x squared n divide by (2n plus one)!
  • ((-1)n)*x2n/(2n+1)!
  • -1n*x2n/2n+1!
  • ((-1)^n)*x²n/(2n+1)!
  • ((-1) to the power of n)*x to the power of 2n/(2n+1)!
  • ((-1)^n)x^2n/(2n+1)!
  • ((-1)n)x2n/(2n+1)!
  • -1nx2n/2n+1!
  • -1^nx^2n/2n+1!
  • ((-1)^n)*x^2n divide by (2n+1)!

  • Similar expressions

  • ((1)^n)*x^2n/(2n+1)!
  • ((-1)^n)*x^2n/(2n-1)!
Sum of series/ ((-1)^n)*x^2n/(2n+1)!

Sum of series ((-1)^n)*x^2n/(2n+1)!


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \        n  2  
  \   (-1) *x *n
  /   ----------
 /    (2*n + 1)!
/___,           
n = 1
$$\sum_{n=1}^{\infty} \frac{n \left(-1\right)^{n} x^{2}}{\left(2 n + 1\right)!}$$ 
Sum((((-1)^n*x^2)*n)/factorial(2*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n \left(-1\right)^{n} x^{2}}{\left(2 n + 1\right)!}$$
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
$$a_{n} = \frac{n x^{2}}{\left(2 n + 1\right)!}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(1 + \lim_{n \to \infty}\left(\frac{n \left|{\frac{\left(2 n + 3\right)!}{\left(2 n + 1\right)!}}\right|}{n + 1}\right)\right)$$
Let's take the limit
we find
$$R = \infty$$
The answer [src] 
 2 /cos(1)   sin(1)\
x *|------ - ------|
   \  2        2   /
$$x^{2} \left(- \frac{\sin{\left(1 \right)}}{2} + \frac{\cos{\left(1 \right)}}{2}\right)$$ 
x^2*(cos(1)/2 - sin(1)/2)

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