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  • Sum of series:
  • 3/(n*(n+2)) 3/(n*(n+2))
  • 3/(n(n+2)) 3/(n(n+2))
  • 1/((3n+1)*(3n+4)) 1/((3n+1)*(3n+4))
  • (sin(2^n))^2/n^2 (sin(2^n))^2/n^2

  • Identical expressions

  • x^n/factorial(n^ one)
  • x to the power of n divide by factorial(n to the power of 1)
  • x to the power of n divide by factorial(n to the power of one)
  • xn/factorial(n1)
  • xn/factorialn1
  • x^n/factorialn^1
  • x^n divide by factorial(n^1)
Sum of series/ x^n/factorial(n^1)

Sum of series x^n/factorial(n^1)


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The solution

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

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