Mister Exam
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-
How to use it?
- Sum of series:
-
sin(pi*n/3)
- cos(x)^2
-
1/(nln^2n)
-
ln(1+1/n^2)
- Limit of the function:
-
sin(pi*n/3)
-
Identical expressions
- sin(pi*n/ three)
- sinus of ( Pi multiply by n divide by 3)
- sinus of ( Pi multiply by n divide by three)
- sin(pin/3)
- sinpin/3
- sin(pi*n divide by 3)
-
Similar expressions
- 1/pi*n((sin(2pin/3)+9sin(4pin/3)-8sin(pin/3))cos(pinx/3)+(cos(2pin/3)+2cos(pin/3)-3cos(4pin/3))sin(pinx/3))
- sin(pi*n)/3
Sum of series sin(pi*n/3)
The solution
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[src]
oo ___ \ ` \ /pi*n\ ) sin|----| / \ 3 / /__, n = 1
$$\sum_{n=1}^{\infty} \sin{\left(\frac{\pi n}{3} \right)}$$
Sum(sin((pi*n)/3), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sin{\left(\frac{\pi n}{3} \right)}$$
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
$$a_{n} = \sin{\left(\frac{\pi n}{3} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi n}{3} \right)}}{\sin{\left(\pi \left(\frac{n}{3} + \frac{1}{3}\right) \right)}}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi n}{3} \right)}}{\sin{\left(\pi \left(\frac{n}{3} + \frac{1}{3}\right) \right)}}}\right|$$
$$\sin{\left(\frac{\pi n}{3} \right)}$$
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
$$a_{n} = \sin{\left(\frac{\pi n}{3} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi n}{3} \right)}}{\sin{\left(\pi \left(\frac{n}{3} + \frac{1}{3}\right) \right)}}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi n}{3} \right)}}{\sin{\left(\pi \left(\frac{n}{3} + \frac{1}{3}\right) \right)}}}\right|$$
False
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ /pi*n\ ) sin|----| / \ 3 / /__, n = 1
$$\sum_{n=1}^{\infty} \sin{\left(\frac{\pi n}{3} \right)}$$
Sum(sin(pi*n/3), (n, 1, oo))
Numerical answer
The series diverges
The graph
Examples of finding the sum of a series
- The Sum of the Power Series
x^n/n
(x-1)^n
- Factorial
1/2^(n!)
n^2/n!
x^n/n!
k!/(n!*(n+k)!)
- Flint Hills Series
csc(n)^2/n^3
- Basel problem
1/n^2
1/n^4
1/n^6
- Harmonic series
1/n
- Grandi's series
(-1)^n
- Alternating series
(-1)^(n + 1)/n
(n + 2)*(-1)^(n - 1)
(3*n - 1)/(-5)^n
(-1)^(n - 1)*n/(6*n - 5)
- Newton–Mercator series
(-1)^(n + 1)/n*x^n
- Exam the series for convergence
(3*n - 1)/(-5)^n