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Sum of series/ (4x)^(2n)

Sum of series (4x)^(2n)


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The solution

You have entered [src] 
  oo          
 ___          
 \  `         
  \        2*n
  /   (4*x)   
 /__,         
n = 0
$$\sum_{n=0}^{\infty} \left(4 x\right)^{2 n}$$ 
Sum((4*x)^(2*n), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\left(4 x\right)^{2 n}$$
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
$$x_{0} = 0$$
,
$$d = 2$$
,
$$c = 4$$
then
$$R^{2} = \frac{\lim_{n \to \infty} 1}{4}$$
Let's take the limit
we find
$$R^{2} = \frac{1}{4}$$
$$R = 0.5$$
The answer [src] 
/       1                | 2|    
|   ---------     for 16*|x | < 1
|           2                    
|   1 - 16*x                     
|                                
|  oo                            
< ___                            
| \  `                           
|  \    2*n  2*n                 
|  /   4   *x        otherwise   
| /__,                           
|n = 0                           
\
$$\begin{cases} \frac{1}{1 - 16 x^{2}} & \text{for}\: 16 \left|{x^{2}}\right| < 1 \\\sum_{n=0}^{\infty} 4^{2 n} x^{2 n} & \text{otherwise} \end{cases}$$ 
Piecewise((1/(1 - 16*x^2), 16*|x^2| < 1), (Sum(4^(2*n)*x^(2*n), (n, 0, oo)), True))

    Examples of finding the sum of a series

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