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(3-2i-5i^2+8i^3)

Sum of series (3-2i-5i^2+8i^3)



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The solution

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  oo                         
 ___                         
 \  `                        
  \   /             2      3\
  /   \3 - 2*i - 5*i  + 8*i /
 /__,                        
i = 1                        
$$\sum_{i=1}^{\infty} \left(8 i^{3} + \left(- 5 i^{2} + \left(3 - 2 i\right)\right)\right)$$
Sum(3 - 2*i - 5*i^2 + 8*i^3, (i, 1, oo))
The radius of convergence of the power series
Given number:
$$8 i^{3} + \left(- 5 i^{2} + \left(3 - 2 i\right)\right)$$
It is a series of species
$$a_{i} \left(c x - x_{0}\right)^{d i}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{i \to \infty} \left|{\frac{a_{i}}{a_{i + 1}}}\right|}{c}$$
In this case
$$a_{i} = 8 i^{3} - 5 i^{2} - 2 i + 3$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{i \to \infty}\left(\frac{\left|{8 i^{3} - 5 i^{2} - 2 i + 3}\right|}{- 2 i + 8 \left(i + 1\right)^{3} - 5 \left(i + 1\right)^{2} + 1}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src]
  oo                         
 ___                         
 \  `                        
  \   /       2            3\
  /   \3 - 5*i  - 2*i + 8*i /
 /__,                        
i = 1                        
$$\sum_{i=1}^{\infty} \left(8 i^{3} - 5 i^{2} - 2 i + 3\right)$$
Sum(3 - 5*i^2 - 2*i + 8*i^3, (i, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series (3-2i-5i^2+8i^3)

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