Mister Exam
Other calculators
-
How to use it?
- Sum of series:
-
1/n
-
sin(pi/n)
- f(x)
-
sqrt(1+5n/n)
- Limit of the function:
-
sin(pi/n)
-
Identical expressions
- sin(pi/n)
- sinus of ( Pi divide by n)
- sinpi/n
- sin(pi divide by n)
-
Similar expressions
- sqrt(n)*sin(pi/n^2)
- arcsin(pi/(n^2+2))
- (sin(pi/n^3))^2n
- sinpi/n
-
Expressions with functions
- Sinus sin
- sin^2(1/4n)
- sin(n^n*factorial(n))/2^n
- sin(π/4)(2n-1)
- sin((2n-1)-x)/(2n-1)^2
- sinn/sqrtn
- Number Pi pi
- pi^(-n)/pi^n+3
- pi^(2*n)/factorial(2*n)
- pi(n)/n
- pi
- pi^n+1/2^2n
Sum of series sin(pi/n)
The solution
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oo ___ \ ` \ /pi\ ) sin|--| / \n / /__, n = 1
$$\sum_{n=1}^{\infty} \sin{\left(\frac{\pi}{n} \right)}$$
Sum(sin(pi/n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sin{\left(\frac{\pi}{n} \right)}$$
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
$$a_{n} = \sin{\left(\frac{\pi}{n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi}{n} \right)}}{\sin{\left(\frac{\pi}{n + 1} \right)}}}\right|$$
Let's take the limit
we find
$$\sin{\left(\frac{\pi}{n} \right)}$$
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
$$a_{n} = \sin{\left(\frac{\pi}{n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi}{n} \right)}}{\sin{\left(\frac{\pi}{n + 1} \right)}}}\right|$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The graph