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  • Sum of series:
  • n^(1/n) n^(1/n)
  • 5^n 5^n
  • 2(1/(n^2+5n+6)) 2(1/(n^2+5n+6))
  • ((-1)^n)*((x^n)/(2n*n!))

  • Identical expressions

  • ((- one)^n)*((x^n)/(2n*n!))
  • (( minus 1) to the power of n) multiply by ((x to the power of n) divide by (2n multiply by n!))
  • (( minus one) to the power of n) multiply by ((x to the power of n) divide by (2n multiply by n!))
  • ((-1)n)*((xn)/(2n*n!))
  • -1n*xn/2n*n!
  • ((-1)^n)((x^n)/(2nn!))
  • ((-1)n)((xn)/(2nn!))
  • -1nxn/2nn!
  • -1^nx^n/2nn!
  • ((-1)^n)*((x^n) divide by (2n*n!))

  • Similar expressions

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

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


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A solução

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

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