Mister Exam
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- Fourier series step by step
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- Product of series
-
How to use it?
- Sum of series:
-
sin(1/n)
-
cos(npi*2)/n^4
-
sqrt(n)*sin(pi/n^2)
-
(-1)^nsin(1/n)
-
Identical expressions
- cos(npi* two)/n^ four
- co sinus of e of (n Pi multiply by 2) divide by n to the power of 4
- co sinus of e of (n Pi multiply by two) divide by n to the power of four
- cos(npi*2)/n4
- cosnpi*2/n4
- cos(npi*2)/n⁴
- cos(npi2)/n^4
- cos(npi2)/n4
- cosnpi2/n4
- cosnpi2/n^4
- cos(npi*2) divide by n^4
Sum of series cos(npi*2)/n^4
The solution
You have entered
[src]
oo ____ \ ` \ cos(n*pi*2) \ ----------- / 4 / n /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\cos{\left(2 \pi n \right)}}{n^{4}}$$
Sum(cos((n*pi)*2)/n^4, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\cos{\left(2 \pi n \right)}}{n^{4}}$$
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
$$a_{n} = \frac{\cos{\left(2 \pi n \right)}}{n^{4}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{4} \left|{\frac{\cos{\left(2 \pi n \right)}}{\cos{\left(\pi \left(2 n + 2\right) \right)}}}\right|}{n^{4}}\right)$$
Let's take the limit
we find
$$\frac{\cos{\left(2 \pi n \right)}}{n^{4}}$$
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
$$a_{n} = \frac{\cos{\left(2 \pi n \right)}}{n^{4}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{4} \left|{\frac{\cos{\left(2 \pi n \right)}}{\cos{\left(\pi \left(2 n + 2\right) \right)}}}\right|}{n^{4}}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
oo ____ \ ` \ cos(2*pi*n) \ ----------- / 4 / n /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\cos{\left(2 \pi n \right)}}{n^{4}}$$
Sum(cos(2*pi*n)/n^4, (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