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  • Sum of series:
  • (1/2)^n (1/2)^n
  • 3 3
  • 0.4 0.4
  • 3^n+5^n/15^n 3^n+5^n/15^n

  • Identical expressions

  • (two x+ one)/(x(x^2- one))
  • (2x plus 1) divide by (x(x squared minus 1))
  • (two x plus one) divide by (x(x squared minus one))
  • (2x+1)/(x(x2-1))
  • 2x+1/xx2-1
  • (2x+1)/(x(x²-1))
  • (2x+1)/(x(x to the power of 2-1))
  • 2x+1/xx^2-1
  • (2x+1) divide by (x(x^2-1))

  • Similar expressions

  • (2x-1)/(x(x^2-1))
  • (2x+1)/(x(x^2+1))
Sum of series/ (2x+1)/(x(x^2-1))

Sum of series (2x+1)/(x(x^2-1))


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The solution

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

False
The answer [src] 
oo*(1 + 2*x)
------------
  /      2\ 
x*\-1 + x /
$$\frac{\infty \left(2 x + 1\right)}{x \left(x^{2} - 1\right)}$$ 
oo*(1 + 2*x)/(x*(-1 + x^2))

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