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(-1)^n/3^n

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  • Sum of series:
  • (n+1)x^n
  • (-1)^(n+1)/(n+1)! (-1)^(n+1)/(n+1)!
  • (-1)^n/3^n (-1)^n/3^n
  • (k²-2) (k²-2)

  • Identical expressions

  • (- one)^n/ three ^n
  • ( minus 1) to the power of n divide by 3 to the power of n
  • ( minus one) to the power of n divide by three to the power of n
  • (-1)n/3n
  • -1n/3n
  • -1^n/3^n
  • (-1)^n divide by 3^n

  • Similar expressions

  • (1)^n/3^n
Sum of series/ (-1)^n/3^n

Sum of series (-1)^n/3^n


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The solution

You have entered [src] 
  oo       
____       
\   `      
 \        n
  \   (-1) 
   )  -----
  /      n 
 /      3  
/___,      
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n}}{3^{n}}$$ 
Sum((-1)^n/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n}}{3^{n}}$$
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
$$a_{n} = \left(-1\right)^{n}$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
-1/4
$$- \frac{1}{4}$$ 
-1/4
Numerical answer [src] 
-0.250000000000000000000000000000
-0.250000000000000000000000000000
The graph
Sum of series (-1)^n/3^n

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