Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
-
How to use it?
- Sum of series:
-
(13^(2n-1))/((2^n)*(n-1)!)
-
(-1)^(n+1)*(n+1)/n
- (-1)^nx^2n+1/(2n+1)!
-
(-1)^n*pi^(2n)/((2n)!)
-
Identical expressions
- (- one)^n*pi^(2n)/((2n)!)
- ( minus 1) to the power of n multiply by Pi to the power of (2n) divide by ((2n)!)
- ( minus one) to the power of n multiply by Pi to the power of (2n) divide by ((2n)!)
- (-1)n*pi(2n)/((2n)!)
- -1n*pi2n/2n!
- (-1)^npi^(2n)/((2n)!)
- (-1)npi(2n)/((2n)!)
- -1npi2n/2n!
- -1^npi^2n/2n!
- (-1)^n*pi^(2n) divide by ((2n)!)
-
Similar expressions
- (1)^n*pi^(2n)/((2n)!)
Sum of series (-1)^n*pi^(2n)/((2n)!)
The solution
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oo ____ \ ` \ n 2*n \ (-1) *pi / ----------- / (2*n)! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n} \pi^{2 n}}{\left(2 n\right)!}$$
Sum(((-1)^n*pi^(2*n))/factorial(2*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n} \pi^{2 n}}{\left(2 n\right)!}$$
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
$$a_{n} = \frac{\left(-1\right)^{n}}{\left(2 n\right)!}$$
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
$$R^{2} = \infty$$
$$R = \infty$$
$$\frac{\left(-1\right)^{n} \pi^{2 n}}{\left(2 n\right)!}$$
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
$$a_{n} = \frac{\left(-1\right)^{n}}{\left(2 n\right)!}$$
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 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