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1/(4n^(2)-1)
Sum of series/ 1/(4n^(2)-1)

Sum of series 1/(4n^(2)-1)


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \       1    
  \   --------
  /      2    
 /    4*n  - 1
/___,         
n = 0
∑n=0∞14n2−1\sum_{n=0}^{\infty} \frac{1}{4 n^{2} - 1} 
Sum(1/(4*n^2 - 1), (n, 0, oo))
The radius of convergence of the power series
Given number:
14n2−1\frac{1}{4 n^{2} - 1}
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=14n2−1a_{n} = \frac{1}{4 n^{2} - 1}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞((4(n+1)2−1)∣14n2−1∣)1 = \lim_{n \to \infty}\left(\left(4 \left(n + 1\right)^{2} - 1\right) \left|{\frac{1}{4 n^{2} - 1}}\right|\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
0.06.00.51.01.52.02.53.03.54.04.55.05.5-1.5-0.5
The answer [src] 
-1/2
−12- \frac{1}{2} 
-1/2
Numerical answer [src] 
-0.500000000000000000000000000000
-0.500000000000000000000000000000
The graph
Sum of series 1/(4n^(2)-1)

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