Mister Exam
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How to use it?
- Sum of series:
- (cos(nx)*3^n)/2^n
- x^(4n)/(4n)!
-
sin(pi*n/4)/ln(n)
-
sin*1/n
-
Identical expressions
- x^(4n)/(4n)!
- x to the power of (4n) divide by (4n)!
- x(4n)/(4n)!
- x4n/4n!
- x^4n/4n!
- x^(4n) divide by (4n)!
Sum of series x^(4n)/(4n)!
The solution
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[src]
oo ____ \ ` \ 4*n \ x / ------ / (4*n)! /___, n = 0
$$\sum_{n=0}^{\infty} \frac{x^{4 n}}{\left(4 n\right)!}$$
Sum(x^(4*n)/factorial(4*n), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{x^{4 n}}{\left(4 n\right)!}$$
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
$$a_{n} = \frac{1}{\left(4 n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 4$$
,
$$c = 1$$
then
$$R^{4} = \lim_{n \to \infty} \left|{\frac{\left(4 n + 4\right)!}{\left(4 n\right)!}}\right|$$
Let's take the limit
we find
$$R^{4} = \infty$$
$$R = \infty$$
$$\frac{x^{4 n}}{\left(4 n\right)!}$$
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
$$a_{n} = \frac{1}{\left(4 n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 4$$
,
$$c = 1$$
then
$$R^{4} = \lim_{n \to \infty} \left|{\frac{\left(4 n + 4\right)!}{\left(4 n\right)!}}\right|$$
Let's take the limit
we find
$$R^{4} = \infty$$
$$R = \infty$$
The answer
[src]
___ ____ ___ /besseli(-1/2, x) besselj(-1/2, x)\
\/ 2 *\/ pi *\/ x *|---------------- + ----------------|
\ 4 4 /
$$\sqrt{2} \sqrt{\pi} \sqrt{x} \left(\frac{I_{- \frac{1}{2}}\left(x\right)}{4} + \frac{J_{- \frac{1}{2}}\left(x\right)}{4}\right)$$
sqrt(2)*sqrt(pi)*sqrt(x)*(besseli(-1/2, x)/4 + besselj(-1/2, x)/4)
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