Sum of series arctgpi/n

The solution

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  oo          
 ___          
 \  `         
  \   atan(pi)
   )  --------
  /      n    
 /__,         
n = 1

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

Sum(atan(pi)/n, (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\operatorname{atan}{\left(\pi \right)}}{n}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(\pi \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n + 1}{n}\right)$$
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

Examples of finding the sum of a series

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```
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```
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```
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```
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```
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```
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```
Alternating series
```
(-1)^(n + 1)/n
```
```
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```
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```
```
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```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Identical expressions

Similar expressions

Sum of series arctgpi/n

The solution

Examples of finding the sum of a series