Mister Exam
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- Product of series
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How to use it?
- Sum of series:
-
5/((5n-2)*(5n+3))
-
sin(pi/n)^n
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(1)/((nlnn)(lnlnn)^2)
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(5^n-2^n)/10^n
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Identical expressions
- five /((5n- two)*(5n+ three))
- 5 divide by ((5n minus 2) multiply by (5n plus 3))
- five divide by ((5n minus two) multiply by (5n plus three))
- 5/((5n-2)(5n+3))
- 5/5n-25n+3
- 5 divide by ((5n-2)*(5n+3))
-
Similar expressions
- 5/(5n-2)(5n+3)
- 5/((5n-2)(5n+3))
- 5/((5n-2)*(5n-3))
- 5/((5n+2)*(5n+3))
Sum of series 5/((5n-2)*(5n+3))
The solution
You have entered
[src]
oo ___ \ ` \ 5 ) ------------------- / (5*n - 2)*(5*n + 3) /__, n = 1
$$\sum_{n=1}^{\infty} \frac{5}{\left(5 n - 2\right) \left(5 n + 3\right)}$$
Sum(5/(((5*n - 2)*(5*n + 3))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{5}{\left(5 n - 2\right) \left(5 n + 3\right)}$$
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
$$a_{n} = \frac{5}{\left(5 n - 2\right) \left(5 n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5 n + 8\right) \left|{\frac{1}{5 n - 2}}\right|\right)$$
Let's take the limit
we find
$$\frac{5}{\left(5 n - 2\right) \left(5 n + 3\right)}$$
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
$$a_{n} = \frac{5}{\left(5 n - 2\right) \left(5 n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5 n + 8\right) \left|{\frac{1}{5 n - 2}}\right|\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
5*Gamma(13/5) ------------- 24*Gamma(8/5)
$$\frac{5 \Gamma\left(\frac{13}{5}\right)}{24 \Gamma\left(\frac{8}{5}\right)}$$
5*gamma(13/5)/(24*gamma(8/5))
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