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  • Sum of series:
  • 1 1
  • sin2n sin2n
  • 2(x-6)^(4n)/(n(16)^n)
  • nx^n/(8n-9)*3^n

  • Identical expressions

  • two (x- six)^(4n)/(n(sixteen)^n)
  • 2(x minus 6) to the power of (4n) divide by (n(16) to the power of n)
  • two (x minus six) to the power of (4n) divide by (n(sixteen) to the power of n)
  • 2(x-6)(4n)/(n(16)n)
  • 2x-64n/n16n
  • 2x-6^4n/n16^n
  • 2(x-6)^(4n) divide by (n(16)^n)

  • Similar expressions

  • 2(x+6)^(4n)/(n(16)^n)
Sum of series/ 2(x-6)^(4n)/(n(16)^n)

Sum of series 2(x-6)^(4n)/(n(16)^n)


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The solution

You have entered [src] 
  oo              
____              
\   `             
 \             4*n
  \   2*(x - 6)   
   )  ------------
  /          n    
 /       n*16     
/___,             
n = 1
$$\sum_{n=1}^{\infty} \frac{2 \left(x - 6\right)^{4 n}}{16^{n} n}$$ 
Sum((2*(x - 6)^(4*n))/((n*16^n)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2 \left(x - 6\right)^{4 n}}{16^{n} n}$$
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
$$a_{n} = \frac{2 \cdot 16^{- n}}{n}$$
and
$$x_{0} = 6$$
,
$$d = 4$$
,
$$c = 1$$
then
$$R^{4} = 6 + \lim_{n \to \infty}\left(\frac{16^{- n} 16^{n + 1} \left(n + 1\right)}{n}\right)$$
Let's take the limit
we find
$$R^{4} = 22$$
$$R = 2.16573677066799$$
The answer [src] 
  //     /            4\                         \
  ||     |    (-6 + x) |                         |
  || -log|1 - ---------|    for And(x > 4, x < 8)|
  ||     \        16   /                         |
  ||                                             |
  ||  oo                                         |
  ||____                                         |
2*|<\   `                                        |
  || \      -n         4*n                       |
  ||  \   16  *(-6 + x)                          |
  ||  /   ----------------        otherwise      |
  || /           n                               |
  ||/___,                                        |
  ||n = 1                                        |
  \\                                             /
$$2 \left(\begin{cases} - \log{\left(1 - \frac{\left(x - 6\right)^{4}}{16} \right)} & \text{for}\: x > 4 \wedge x < 8 \\\sum_{n=1}^{\infty} \frac{16^{- n} \left(x - 6\right)^{4 n}}{n} & \text{otherwise} \end{cases}\right)$$ 
2*Piecewise((-log(1 - (-6 + x)^4/16), (x > 4)∧(x < 8)), (Sum(16^(-n)*(-6 + x)^(4*n)/n, (n, 1, oo)), True))

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