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  • 1/4^x
  • Derivative of:
  • 1/4^x

  • Identical expressions

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

Sum of series 1/4^x


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The solution

You have entered [src] 
  oo     
 ___     
 \  `    
  \    -x
  /   4  
 /__,    
n = 1
∑n=1∞(14)x\sum_{n=1}^{\infty} \left(\frac{1}{4}\right)^{x} 
Sum((1/4)^x, (n, 1, oo))
The radius of convergence of the power series
Given number:
(14)x\left(\frac{1}{4}\right)^{x}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=(14)xa_{n} = \left(\frac{1}{4}\right)^{x}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞11 = \lim_{n \to \infty} 1
Let's take the limit
we find
True

False
The answer [src] 
    -x
oo*4
∞4−x\infty 4^{- x} 
oo*4^(-x)

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