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

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


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The solution

You have entered [src] 
  oo                 
 ___                 
 \  `                
  \   / n      1    \
   )  |x  - --------|
  /   \     (n - 1)!/
 /__,                
n = 1
$$\sum_{n=1}^{\infty} \left(x^{n} - \frac{1}{\left(n - 1\right)!}\right)$$ 
Sum(x^n - 1/factorial(n - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$x^{n} - \frac{1}{\left(n - 1\right)!}$$
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
$$a_{n} = x^{n} - \frac{1}{\left(n - 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{x^{n} - \frac{1}{\left(n - 1\right)!}}{x^{n + 1} - \frac{1}{n!}}}\right|$$
Let's take the limit
we find
$$1 = \frac{\operatorname{sign}{\left(\frac{1}{x} \right)}}{x}$$
False
The answer [src] 
     //   x                 \
     || -----    for |x| < 1|
     || 1 - x               |
     ||                     |
     ||  oo                 |
-E + |< ___                 |
     || \  `                |
     ||  \    n             |
     ||  /   x    otherwise |
     || /__,                |
     \\n = 1                /
$$\begin{cases} \frac{x}{1 - x} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=1}^{\infty} x^{n} & \text{otherwise} \end{cases} - e$$ 
-E + Piecewise((x/(1 - x), |x| < 1), (Sum(x^n, (n, 1, oo)), True))

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