Mister Exam
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How to use it?
- Sum of series:
-
sqrt((n+4)/((n^4)+4))
- tg^(2)*(6/(2n)^(1/3))*(x-24)^n
-
(4*9^n-2^n)/18^n
- cos(kx)
-
Identical expressions
- (eight /((two *n+ one)*(two *n+ one)* three . one thousand, four hundred and sixteen * three . one thousand, four hundred and sixteen))*exp(-(two *n+ one)* zero . one hundred and ninety-seven * three . one thousand, four hundred and sixteen * three . one thousand, four hundred and sixteen * zero . twenty-five)
- (8 divide by ((2 multiply by n plus 1) multiply by (2 multiply by n plus 1) multiply by 3.1416 multiply by 3.1416)) multiply by exponent of ( minus (2 multiply by n plus 1) multiply by 0.197 multiply by 3.1416 multiply by 3.1416 multiply by 0.25)
- (eight divide by ((two multiply by n plus one) multiply by (two multiply by n plus one) multiply by three . one thousand, four hundred and sixteen multiply by three . one thousand, four hundred and sixteen)) multiply by exponent of ( minus (two multiply by n plus one) multiply by zero . one hundred and ninety minus seven multiply by three . one thousand, four hundred and sixteen multiply by three . one thousand, four hundred and sixteen multiply by zero . twenty minus five)
- (8/((2n+1)(2n+1)3.14163.1416))exp(-(2n+1)0.1973.14163.14160.25)
- 8/2n+12n+13.14163.1416exp-2n+10.1973.14163.14160.25
- (8 divide by ((2*n+1)*(2*n+1)*3.1416*3.1416))*exp(-(2*n+1)*0.197*3.1416*3.1416*0.25)
-
Similar expressions
- (8/((2*n+1)*(2*n+1)*3.1416*3.1416))*exp(-(2*n-1)*0.197*3.1416*3.1416*0.25)
- (8/((2*n+1)*(2*n+1)*3.1416*3.1416))*exp(-(2n+1)*0.197*3.1416*3.1416*0.25)
- (8/((2*n+1)*(2*n+1)*3.1416*3.1416))*exp((2*n+1)*0.197*3.1416*3.1416*0.25)
- (8/((2*n-1)*(2*n+1)*3.1416*3.1416))*exp(-(2*n+1)*0.197*3.1416*3.1416*0.25)
- (8/((2*n+1)*(2*n-1)*3.1416*3.1416))*exp(-(2*n+1)*0.197*3.1416*3.1416*0.25)
Sum of series (8/((2*n+1)*(2*n+1)*3.1416*3.1416))*exp(-(2*n+1)*0.197*3.1416*3.1416*0.25)
The solution
You have entered
[src]
oo
_____
\ `
\ (-2*n - 1)*197
\ --------------*3.1416*3.1416
\ 1000
) ----------------------------
/ 8 4
/ ---------------------------------*e
/ (2*n + 1)*(2*n + 1)*3.1416*3.1416
/____,
n = 0
$$\sum_{n=0}^{\infty} \frac{8}{3.1416 \cdot 3.1416 \left(2 n + 1\right) \left(2 n + 1\right)} e^{\frac{3.1416 \cdot 3.1416 \frac{197 \left(- 2 n - 1\right)}{1000}}{4}}$$
Sum((8/(((((2*n + 1)*(2*n + 1))*3.1416)*3.1416)))*exp(((((-2*n - 1)*197/1000)*3.1416)*3.1416)/4), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{8}{3.1416 \cdot 3.1416 \left(2 n + 1\right) \left(2 n + 1\right)} e^{\frac{3.1416 \cdot 3.1416 \frac{197 \left(- 2 n - 1\right)}{1000}}{4}}$$
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
$$a_{n} = \frac{0.498524174155581}{\left(2 n + 1\right)^{2}}$$
and
$$x_{0} = - e$$
,
$$d = -0.97216058016$$
,
$$c = 0$$
then
$$R^{-0.97216058016} = \tilde{\infty} \left(- e + \lim_{n \to \infty}\left(\frac{1 \left(2 n + 3\right)^{2}}{\left(2 n + 1\right)^{2}}\right)\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{8}{3.1416 \cdot 3.1416 \left(2 n + 1\right) \left(2 n + 1\right)} e^{\frac{3.1416 \cdot 3.1416 \frac{197 \left(- 2 n - 1\right)}{1000}}{4}}$$
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
$$a_{n} = \frac{0.498524174155581}{\left(2 n + 1\right)^{2}}$$
and
$$x_{0} = - e$$
,
$$d = -0.97216058016$$
,
$$c = 0$$
then
$$R^{-0.97216058016} = \tilde{\infty} \left(- e + \lim_{n \to \infty}\left(\frac{1 \left(2 n + 3\right)^{2}}{\left(2 n + 1\right)^{2}}\right)\right)$$
Let's take the limit
we find
False
$$R = 0$$
The rate of convergence of the power series
The answer
[src]
0.405282839111986*polylog(2, 0.615032424328521) - 0.405282839111986*polylog(2, -0.615032424328521)
$$- 0.405282839111986 \operatorname{Li}_{2}\left(-0.615032424328521\right) + 0.405282839111986 \operatorname{Li}_{2}\left(0.615032424328521\right)$$
0.405282839111986*polylog(2, 0.615032424328521) - 0.405282839111986*polylog(2, -0.615032424328521)
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