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(-12/(9n-15)+4/(3n+1))
Sum of series/ (-12/(9n-15)+4/(3n+1))

Sum of series (-12/(9n-15)+4/(3n+1))


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The solution

You have entered [src] 
  oo                        
 ___                        
 \  `                       
  \   /     12         4   \
   )  |- -------- + -------|
  /   \  9*n - 15   3*n + 1/
 /__,                       
n = 1
$$\sum_{n=1}^{\infty} \left(- \frac{12}{9 n - 15} + \frac{4}{3 n + 1}\right)$$ 
Sum(-12/(9*n - 15) + 4/(3*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$- \frac{12}{9 n - 15} + \frac{4}{3 n + 1}$$
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
$$a_{n} = - \frac{12}{9 n - 15} + \frac{4}{3 n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 4\right) \left|{9 n - 6}\right|}{\left(3 n + 1\right) \left|{9 n - 15}\right|}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
-3*Gamma(7/3)
-------------
 2*Gamma(4/3)
$$- \frac{3 \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{4}{3}\right)}$$ 
-3*gamma(7/3)/(2*gamma(4/3))
Numerical answer [src] 
-2.00000000000000000000000000000
-2.00000000000000000000000000000
The graph
Sum of series (-12/(9n-15)+4/(3n+1))

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