Sum of series sin(pi/n)^n

The solution

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  oo          
 ___          
 \  `         
  \      n/pi\
   )  sin |--|
  /       \n /
 /__,         
n = 1

$$\sum_{n=1}^{\infty} \sin^{n}{\left(\frac{\pi}{n} \right)}$$

Sum(sin(pi/n)^n, (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\sin^{n}{\left(\frac{\pi}{n} \right)}$$
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
$$a_{n} = \sin^{n}{\left(\frac{\pi}{n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\sin^{n}{\left(\frac{\pi}{n} \right)}}\right|}{\left|{\sin^{n + 1}{\left(\frac{\pi}{n + 1} \right)}}\right|}\right)$$
Let's take the limit
we find

False

False

The rate of convergence of the power series

Numerical answer [src]

1.98873236754610398181308175190

1.98873236754610398181308175190

The graph

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```
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```
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```
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```
Exam the series for convergence
```
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```

Other calculators

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Identical expressions

Sum of series sin(pi/n)^n

The solution

Examples of finding the sum of a series