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pi/3^(2n+1)/(2n+1)!

Sum of series pi/3^(2n+1)/(2n+1)!



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The solution

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

False
The rate of convergence of the power series
The answer [src]
pi*(-1 + 3*sinh(1/3))
---------------------
          3          
$$\frac{\pi \left(-1 + 3 \sinh{\left(\frac{1}{3} \right)}\right)}{3}$$
pi*(-1 + 3*sinh(1/3))/3
Numerical answer [src]
0.0195005690751080947647492766508
0.0195005690751080947647492766508
The graph
Sum of series pi/3^(2n+1)/(2n+1)!

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