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24/(9n^2-12n+5)
Sum of series/ 24/(9n^2-12n+5)

Sum of series 24/(9n^2-12n+5)


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The solution

You have entered [src] 
  oo                 
____                 
\   `                
 \           24      
  \   ---------------
  /      2           
 /    9*n  - 12*n + 5
/___,                
n = 1
∑n=1∞24(9n2−12n)+5\sum_{n=1}^{\infty} \frac{24}{\left(9 n^{2} - 12 n\right) + 5} 
Sum(24/(9*n^2 - 12*n + 5), (n, 1, oo))
The radius of convergence of the power series
Given number:
24(9n2−12n)+5\frac{24}{\left(9 n^{2} - 12 n\right) + 5}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=249n2−12n+5a_{n} = \frac{24}{9 n^{2} - 12 n + 5}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞((−12n+9(n+1)2−7)∣19n2−12n+5∣)1 = \lim_{n \to \infty}\left(\left(- 12 n + 9 \left(n + 1\right)^{2} - 7\right) \left|{\frac{1}{9 n^{2} - 12 n + 5}}\right|\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.510.015.0
Numerical answer [src] 
14.8194952335788599394851486518
14.8194952335788599394851486518
The graph
Sum of series 24/(9n^2-12n+5)

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