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1/4^n

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  • Sum of series:
  • 1/4^n 1/4^n
  • (2n-1)/2^n (2n-1)/2^n
  • sinn/sqrtn sinn/sqrtn
  • 64 64

  • Identical expressions

  • one / four ^n
  • 1 divide by 4 to the power of n
  • one divide by four to the power of n
  • 1/4n
  • 1 divide by 4^n
Sum of series/ 1/4^n

Sum of series 1/4^n


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The solution

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

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