Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
sqrt((n+4)/((n^4)+4))
- cos(kx)
- ((-1)^(n-1))*(5^n+2)*x^n
-
sqrt(n+1)-sqrt(n)/sqrt(n^2+n)
-
Identical expressions
- factorial(n)*factorial(two *n+ two)/factorial(three *n)
- factorial(n) multiply by factorial(2 multiply by n plus 2) divide by factorial(3 multiply by n)
- factorial(n) multiply by factorial(two multiply by n plus two) divide by factorial(three multiply by n)
- factorial(n)factorial(2n+2)/factorial(3n)
- factorialnfactorial2n+2/factorial3n
- factorial(n)*factorial(2*n+2) divide by factorial(3*n)
-
Similar expressions
- factorial(n)*factorial(2*n-2)/factorial(3*n)
Sum of series factorial(n)*factorial(2*n+2)/factorial(3*n)
The solution
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[src]
oo ___ \ ` \ n!*(2*n + 2)! ) ------------- / (3*n)! /__, n = 1
$$\sum_{n=1}^{\infty} \frac{n! \left(2 n + 2\right)!}{\left(3 n\right)!}$$
Sum((factorial(n)*factorial(2*n + 2))/factorial(3*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n! \left(2 n + 2\right)!}{\left(3 n\right)!}$$
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
$$a_{n} = \frac{n! \left(2 n + 2\right)!}{\left(3 n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{n! \left(2 n + 2\right)! \left(3 n + 3\right)!}{\left(3 n\right)! \left(n + 1\right)! \left(2 n + 4\right)!}}\right|$$
Let's take the limit
we find
$$\frac{n! \left(2 n + 2\right)!}{\left(3 n\right)!}$$
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
$$a_{n} = \frac{n! \left(2 n + 2\right)!}{\left(3 n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{n! \left(2 n + 2\right)! \left(3 n + 3\right)!}{\left(3 n\right)! \left(n + 1\right)! \left(2 n + 4\right)!}}\right|$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
_ |_ /1, 5/2, 3 | \ 4* | | | 4/27| 3 2 \ 4/3, 5/3 | /
$$4 {{}_{3}F_{2}\left(\begin{matrix} 1, \frac{5}{2}, 3 \\ \frac{4}{3}, \frac{5}{3} \end{matrix}\middle| {\frac{4}{27}} \right)}$$
4*hyper((1, 5/2, 3), (4/3, 5/3), 4/27)
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