Sum of series sen((npi)/2)

The solution

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  oo           
 ___           
 \  `          
  \      /n*pi\
   )  sin|----|
  /      \ 2  /
 /__,          
n = 1

$$\sum_{n=1}^{\infty} \sin{\left(\frac{\pi n}{2} \right)}$$

Sum(sin((n*pi)/2), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\sin{\left(\frac{\pi n}{2} \right)}$$
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
$$a_{n} = \sin{\left(\frac{\pi n}{2} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi n}{2} \right)}}{\sin{\left(\pi \left(\frac{n}{2} + \frac{1}{2}\right) \right)}}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi n}{2} \right)}}{\sin{\left(\pi \left(\frac{n}{2} + \frac{1}{2}\right) \right)}}}\right|$$

False

The rate of convergence of the power series

The answer [src]

  oo           
 ___           
 \  `          
  \      /pi*n\
   )  sin|----|
  /      \ 2  /
 /__,          
n = 1

$$\sum_{n=1}^{\infty} \sin{\left(\frac{\pi n}{2} \right)}$$

Sum(sin(pi*n/2), (n, 1, 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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(-1)^(n + 1)/n
```
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```
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```
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Newton–Mercator series
```
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```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Sum of series sen((npi)/2)

The solution

Examples of finding the sum of a series