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

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

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  oo            
____            
\   `           
 \     3        
  \   n  + n + 5
  /   ----------
 /      n + 6   
/___,           
n = 1

\sum_{n=1}^{\infty} \frac{\left(n^{3} + n\right) + 5}{n + 6}

Sum((n^3 + n + 5)/(n + 6), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{\left(n^{3} + n\right) + 5}{n + 6}

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{n^{3} + n + 5}{n + 6}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(n + 7\right) \left(n^{3} + n + 5\right)}{\left(n + 6\right) \left(n + \left(n + 1\right)^{3} + 6\right)}\right)

True

False

The rate of convergence of the power series

The answer [src]

oo

\infty

oo

Numerical answer

The series diverges

The graph