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1/(n*(n+3))
Sum of series/ 1/(n*(n+3))

Sum of series 1/(n*(n+3))


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The solution

You have entered [src] 
  oo           
 ___           
 \  `          
  \       1    
   )  ---------
  /   n*(n + 3)
 /__,          
n = 1
∑n=1∞1n(n+3)\sum_{n=1}^{\infty} \frac{1}{n \left(n + 3\right)} 
Sum(1/(n*(n + 3)), (n, 1, oo))
The radius of convergence of the power series
Given number:
1n(n+3)\frac{1}{n \left(n + 3\right)}
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=1n(n+3)a_{n} = \frac{1}{n \left(n + 3\right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞((n+1)(n+4)n(n+3))1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(n + 4\right)}{n \left(n + 3\right)}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.000.75
The answer [src] 
          0               0  
  -1 + 3*e        -5 + 2*e   
------------- + -------------
  /        0\     /        0\
2*\-6 + 6*e /   3*\-6 + 6*e /
−5+2e03(−6+6e0)+−1+3e02(−6+6e0)\frac{-5 + 2 e^{0}}{3 \left(-6 + 6 e^{0}\right)} + \frac{-1 + 3 e^{0}}{2 \left(-6 + 6 e^{0}\right)} 
(-1 + 3*exp_polar(0))/(2*(-6 + 6*exp_polar(0))) + (-5 + 2*exp_polar(0))/(3*(-6 + 6*exp_polar(0)))
Numerical answer [src] 
0.611111111111111111111111111111
0.611111111111111111111111111111
The graph
Sum of series 1/(n*(n+3))

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