Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
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How to use it?
- Sum of series:
-
1/(n(n+1)(n+2))
-
1/3^n
-
1/(3n-2)(3n+1)
- (x-1)^n/5^n
-
Identical expressions
- pi^(two *n)/factorial(two *n)
- Pi to the power of (2 multiply by n) divide by factorial(2 multiply by n)
- Pi to the power of (two multiply by n) divide by factorial(two multiply by n)
- pi(2*n)/factorial(2*n)
- pi2*n/factorial2*n
- pi^(2n)/factorial(2n)
- pi(2n)/factorial(2n)
- pi2n/factorial2n
- pi^2n/factorial2n
- pi^(2*n) divide by factorial(2*n)
Sum of series pi^(2*n)/factorial(2*n)
The solution
You have entered
[src]
oo ____ \ ` \ 2*n \ pi / ------ / (2*n)! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\pi^{2 n}}{\left(2 n\right)!}$$
Sum(pi^(2*n)/factorial(2*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\pi^{2 n}}{\left(2 n\right)!}$$
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
$$a_{n} = \frac{1}{\left(2 n\right)!}$$
and
$$x_{0} = - \pi$$
,
$$d = 2$$
,
$$c = 0$$
then
$$R^{2} = \tilde{\infty} \left(- \pi + \lim_{n \to \infty} \left|{\frac{\left(2 n + 2\right)!}{\left(2 n\right)!}}\right|\right)$$
Let's take the limit
we find
$$R^{2} = \infty$$
$$R = \infty$$
$$\frac{\pi^{2 n}}{\left(2 n\right)!}$$
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
$$a_{n} = \frac{1}{\left(2 n\right)!}$$
and
$$x_{0} = - \pi$$
,
$$d = 2$$
,
$$c = 0$$
then
$$R^{2} = \tilde{\infty} \left(- \pi + \lim_{n \to \infty} \left|{\frac{\left(2 n + 2\right)!}{\left(2 n\right)!}}\right|\right)$$
Let's take the limit
we find
$$R^{2} = \infty$$
$$R = \infty$$
The rate of convergence of the power series
The answer
[src]
2 / 2 2*cosh(pi)\
pi *|- --- + ----------|
| 2 2 |
\ pi pi /
------------------------
2
$$\frac{\pi^{2} \left(- \frac{2}{\pi^{2}} + \frac{2 \cosh{\left(\pi \right)}}{\pi^{2}}\right)}{2}$$
pi^2*(-2/pi^2 + 2*cosh(pi)/pi^2)/2
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