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((-3)^(n+1))/(2^(3n))

Sum of series ((-3)^(n+1))/(2^(3n))



=

The solution

You have entered [src]
  oo           
____           
\   `          
 \        n + 1
  \   (-3)     
   )  ---------
  /       3*n  
 /       2     
/___,          
n = 2          
$$\sum_{n=2}^{\infty} \frac{\left(-3\right)^{n + 1}}{2^{3 n}}$$
Sum((-3)^(n + 1)/2^(3*n), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-3\right)^{n + 1}}{2^{3 n}}$$
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
$$a_{n} = \left(-3\right)^{n + 1}$$
and
$$x_{0} = -2$$
,
$$d = -3$$
,
$$c = 0$$
then
$$\frac{1}{R^{3}} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(3^{- n - 2} \cdot 3^{n + 1}\right)\right)$$
Let's take the limit
we find
False

False

$$R = 0$$
The rate of convergence of the power series
The answer [src]
-27 
----
 88 
$$- \frac{27}{88}$$
-27/88
Numerical answer [src]
-0.306818181818181818181818181818
-0.306818181818181818181818181818
The graph
Sum of series ((-3)^(n+1))/(2^(3n))

    Examples of finding the sum of a series