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:
- x^n/sqrt(n+1)
-
(1000^n)/n!
- ((-1)^n*(3*cos(n*x)-pi*n*sin(n*x))-3*cos(n*x+pi*n/3))/n^2
-
2n
-
Identical expressions
- six /((three *n- two)(three *n+ four))
- 6 divide by ((3 multiply by n minus 2)(3 multiply by n plus 4))
- six divide by ((three multiply by n minus two)(three multiply by n plus four))
- 6/((3n-2)(3n+4))
- 6/3n-23n+4
- 6 divide by ((3*n-2)(3*n+4))
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Similar expressions
- 6/((3*n+2)(3*n+4))
- 6/((3*n-2)(3*n-4))
Sum of series 6/((3*n-2)(3*n+4))
The solution
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[src]
oo ___ \ ` \ 6 ) ------------------- / (3*n - 2)*(3*n + 4) /__, n = 1
$$\sum_{n=1}^{\infty} \frac{6}{\left(3 n - 2\right) \left(3 n + 4\right)}$$
Sum(6/(((3*n - 2)*(3*n + 4))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{6}{\left(3 n - 2\right) \left(3 n + 4\right)}$$
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
$$a_{n} = \frac{6}{\left(3 n - 2\right) \left(3 n + 4\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 1\right) \left(3 n + 7\right) \left|{\frac{1}{3 n - 2}}\right|}{3 n + 4}\right)$$
Let's take the limit
we find
$$\frac{6}{\left(3 n - 2\right) \left(3 n + 4\right)}$$
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
$$a_{n} = \frac{6}{\left(3 n - 2\right) \left(3 n + 4\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 1\right) \left(3 n + 7\right) \left|{\frac{1}{3 n - 2}}\right|}{3 n + 4}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
15*Gamma(10/3) -------------- 28*Gamma(7/3)
$$\frac{15 \Gamma\left(\frac{10}{3}\right)}{28 \Gamma\left(\frac{7}{3}\right)}$$
15*gamma(10/3)/(28*gamma(7/3))
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