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1/(2n+1)*2^2n+1

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  • Sum of series:
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  • k*x+b
  • cos(n*p)/4
  • 1/(2n+1)*2^2n+1 1/(2n+1)*2^2n+1

  • Identical expressions

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  • 1/(2n+1)*2²n+1
  • 1/(2n+1)*2 to the power of 2n+1
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  • Similar expressions

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

Sum of series 1/(2n+1)*2^2n+1


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The solution

You have entered [src] 
  oo                 
 ___                 
 \  `                
  \   /   4         \
   )  |-------*n + 1|
  /   \2*n + 1      /
 /__,                
n = 1
$$\sum_{n=1}^{\infty} \left(n \frac{4}{2 n + 1} + 1\right)$$ 
Sum((4/(2*n + 1))*n + 1, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n \frac{4}{2 n + 1} + 1$$
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
$$a_{n} = \frac{4 n}{2 n + 1} + 1$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\frac{4 n}{2 n + 1} + 1}{\frac{4 \left(n + 1\right)}{2 n + 3} + 1}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
oo
$$\infty$$ 
oo
Numerical answer
The series diverges
The graph
Sum of series 1/(2n+1)*2^2n+1

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