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2*(1/n^3+5n+6)
Sum of series/ 2*(1/n^3+5n+6)

Sum of series 2*(1/n^3+5n+6)


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The solution

You have entered [src] 
  oo                  
____                  
\   `                 
 \      /1           \
  \   2*|-- + 5*n + 6|
  /     | 3          |
 /      \n           /
/___,                 
n = 0
$$\sum_{n=0}^{\infty} 2 \left(\left(5 n + \frac{1}{n^{3}}\right) + 6\right)$$ 
Sum(2*(1/(n^3) + 5*n + 6), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$2 \left(\left(5 n + \frac{1}{n^{3}}\right) + 6\right)$$
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
$$a_{n} = 10 n + 12 + \frac{2}{n^{3}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{10 n + 12 + \frac{2}{n^{3}}}{10 n + 22 + \frac{2}{\left(n + 1\right)^{3}}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo                  
____                  
\   `                 
 \    /     2        \
  \   |12 + -- + 10*n|
  /   |      3       |
 /    \     n        /
/___,                 
n = 0
$$\sum_{n=0}^{\infty} \left(10 n + 12 + \frac{2}{n^{3}}\right)$$ 
Sum(12 + 2/n^3 + 10*n, (n, 0, oo))
Numerical answer [src] 
Sum(2*(1/(n^3) + 5*n + 6), (n, 0, oo))
Sum(2*(1/(n^3) + 5*n + 6), (n, 0, oo))
The graph
Sum of series 2*(1/n^3+5n+6)

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