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(3^n-1)/2^2n-5
Sum of series/ (3^n-1)/2^2n-5

Sum of series (3^n-1)/2^2n-5


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The solution

You have entered [src] 
  oo                
____                
\   `               
 \    / n          \
  \   |3  - 1      |
  /   |------*n - 5|
 /    \  4         /
/___,               
n = 1
$$\sum_{n=1}^{\infty} \left(n \frac{3^{n} - 1}{4} - 5\right)$$ 
Sum(((3^n - 1)/4)*n - 5, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n \frac{3^{n} - 1}{4} - 5$$
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
$$a_{n} = n \left(\frac{3^{n}}{4} - \frac{1}{4}\right) - 5$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{n \left(\frac{3^{n}}{4} - \frac{1}{4}\right) - 5}{\left(\frac{3^{n + 1}}{4} - \frac{1}{4}\right) \left(n + 1\right) - 5}}\right|$$
Let's take the limit
we find
False

False

False
The rate of convergence of the power series
The answer [src] 
  oo                     
____                     
\   `                    
 \    /       /       n\\
  \   |       |  1   3 ||
  /   |-5 + n*|- - + --||
 /    \       \  4   4 //
/___,                    
n = 1
$$\sum_{n=1}^{\infty} \left(n \left(\frac{3^{n}}{4} - \frac{1}{4}\right) - 5\right)$$ 
Sum(-5 + n*(-1/4 + 3^n/4), (n, 1, oo))
The graph
Sum of series (3^n-1)/2^2n-5

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