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4^(n-1)/8^n

  • How to use it?

  • Sum of series:
  • 1/(n(n+1)) 1/(n(n+1))
  • 1/x^n
  • (3^(n+3))/(4^(n-1)*5^n) (3^(n+3))/(4^(n-1)*5^n)
  • 4^(n-1)/8^n 4^(n-1)/8^n

  • Identical expressions

  • four ^(n- one)/ eight ^n
  • 4 to the power of (n minus 1) divide by 8 to the power of n
  • four to the power of (n minus one) divide by eight to the power of n
  • 4(n-1)/8n
  • 4n-1/8n
  • 4^n-1/8^n
  • 4^(n-1) divide by 8^n

  • Similar expressions

  • 4^(n+1)/8^n
Sum of series/ 4^(n-1)/8^n

Sum of series 4^(n-1)/8^n


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The solution

You have entered [src] 
  oo        
____        
\   `       
 \     n - 1
  \   4     
   )  ------
  /      n  
 /      8   
/___,       
n = 1
$$\sum_{n=1}^{\infty} \frac{4^{n - 1}}{8^{n}}$$ 
Sum(4^(n - 1)/8^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{4^{n - 1}}{8^{n}}$$
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
$$a_{n} = 4^{n - 1}$$
and
$$x_{0} = -8$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-8 + \lim_{n \to \infty}\left(4^{- n} 4^{n - 1}\right)\right)$$
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 4^(n-1)/8^n

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