Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
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How to use it?
- Sum of series:
-
n^2*sin(5/(3^n))
-
n*2^n
-
n^3
- t^k*(-1)^k*a^(2*k)*cos(a*x)*e^(a*z)/factorial(k)
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Identical expressions
- (cos(n*pi/ three)- one)cos(n*pi)/n^ two
- ( co sinus of e of (n multiply by Pi divide by 3) minus 1) co sinus of e of (n multiply by Pi ) divide by n squared
- ( co sinus of e of (n multiply by Pi divide by three) minus one) co sinus of e of (n multiply by Pi ) divide by n to the power of two
- (cos(n*pi/3)-1)cos(n*pi)/n2
- cosn*pi/3-1cosn*pi/n2
- (cos(n*pi/3)-1)cos(n*pi)/n²
- (cos(n*pi/3)-1)cos(n*pi)/n to the power of 2
- (cos(npi/3)-1)cos(npi)/n^2
- (cos(npi/3)-1)cos(npi)/n2
- cosnpi/3-1cosnpi/n2
- cosnpi/3-1cosnpi/n^2
- (cos(n*pi divide by 3)-1)cos(n*pi) divide by n^2
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Similar expressions
- (cos(n*pi/3)+1)cos(n*pi)/n^2
Sum of series (cos(n*pi/3)-1)cos(n*pi)/n^2
The solution
You have entered
[src]
oo _____ \ ` \ / /n*pi\ \ \ |cos|----| - 1|*cos(n*pi) \ \ \ 3 / / / ------------------------- / 2 / n /____, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(\cos{\left(\frac{\pi n}{3} \right)} - 1\right) \cos{\left(\pi n \right)}}{n^{2}}$$
Sum(((cos((n*pi)/3) - 1)*cos(n*pi))/n^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(\cos{\left(\frac{\pi n}{3} \right)} - 1\right) \cos{\left(\pi n \right)}}{n^{2}}$$
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
$$a_{n} = \frac{\left(\cos{\left(\frac{\pi n}{3} \right)} - 1\right) \cos{\left(\pi n \right)}}{n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2} \left|{\frac{\left(\cos{\left(\frac{\pi n}{3} \right)} - 1\right) \cos{\left(\pi n \right)}}{\left(\cos{\left(\pi \left(\frac{n}{3} + \frac{1}{3}\right) \right)} - 1\right) \cos{\left(\pi \left(n + 1\right) \right)}}}\right|}{n^{2}}\right)$$
Let's take the limit
we find
$$\frac{\left(\cos{\left(\frac{\pi n}{3} \right)} - 1\right) \cos{\left(\pi n \right)}}{n^{2}}$$
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
$$a_{n} = \frac{\left(\cos{\left(\frac{\pi n}{3} \right)} - 1\right) \cos{\left(\pi n \right)}}{n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2} \left|{\frac{\left(\cos{\left(\frac{\pi n}{3} \right)} - 1\right) \cos{\left(\pi n \right)}}{\left(\cos{\left(\pi \left(\frac{n}{3} + \frac{1}{3}\right) \right)} - 1\right) \cos{\left(\pi \left(n + 1\right) \right)}}}\right|}{n^{2}}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
oo _____ \ ` \ / /pi*n\\ \ |-1 + cos|----||*cos(pi*n) \ \ \ 3 // / -------------------------- / 2 / n /____, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(\cos{\left(\frac{\pi n}{3} \right)} - 1\right) \cos{\left(\pi n \right)}}{n^{2}}$$
Sum((-1 + cos(pi*n/3))*cos(pi*n)/n^2, (n, 1, oo))
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