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((-1)^(n+1))/(2^n)

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



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The solution

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

$$R = 0$$
The rate of convergence of the power series
The answer [src]
1/3
$$\frac{1}{3}$$
1/3
Numerical answer [src]
0.333333333333333333333333333333
0.333333333333333333333333333333
The graph
Sum of series ((-1)^(n+1))/(2^n)

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