Mister Exam

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

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  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)

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