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Sum of series/ n²/12a+n-144a²/12a+n

Sum of series n²/12a+n-144a²/12a+n


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The solution

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  oo                           
____                           
\   `                          
 \    / 2              2      \
  \   |n          144*a       |
  /   |--*a + n - ------*a + n|
 /    \12           12        /
/___,                          
n = 1
$$\sum_{n=1}^{\infty} \left(n + \left(- a \frac{144 a^{2}}{12} + \left(a \frac{n^{2}}{12} + n\right)\right)\right)$$ 
Sum((n^2/12)*a + n - (144*a^2)/12*a + n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n + \left(- a \frac{144 a^{2}}{12} + \left(a \frac{n^{2}}{12} + n\right)\right)$$
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
$$a_{n} = - 12 a^{3} + \frac{a n^{2}}{12} + 2 n$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{- 12 a^{3} + \frac{a n^{2}}{12} + 2 n}{- 12 a^{3} + \frac{a \left(n + 1\right)^{2}}{12} + 2 n + 2}}\right|$$
Let's take the limit
we find
True

False
The answer [src] 
                3
oo + oo*a - oo*a
$$- \infty a^{3} + \infty a + \infty$$ 
oo + oo*a - oo*a^3

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