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

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


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The solution

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

False
The answer [src] 
  oo                      
____                      
\   `                     
 \        1 + n           
  \   (-1)     *sin(2*n*x)
  /   --------------------
 /             n          
/___,                     
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1} \sin{\left(2 n x \right)}}{n}$$ 
Sum((-1)^(1 + n)*sin(2*n*x)/n, (n, 1, oo))

    Examples of finding the sum of a series

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