Mister Exam
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- Product of series
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How to use it?
- Sum of series:
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1/(n*(n+2))
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cos(2*n)/n^2
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1/((2n-1)(2n)(2n+1))
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-1
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Identical expressions
- one /((2n- one)(2n)(2n+ one))
- 1 divide by ((2n minus 1)(2n)(2n plus 1))
- one divide by ((2n minus one)(2n)(2n plus one))
- 1/2n-12n2n+1
- 1 divide by ((2n-1)(2n)(2n+1))
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Similar expressions
- 1/((2n-1)(2n)(2n-1))
- 1/((2n+1)(2n)(2n+1))
- 1/(2n-1)(2n)(2n+1)
Sum of series 1/((2n-1)(2n)(2n+1))
The solution
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oo ___ \ ` \ 1 ) ----------------------- / (2*n - 1)*2*n*(2*n + 1) /__, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{2 n \left(2 n - 1\right) \left(2 n + 1\right)}$$
Sum(1/(((2*n - 1)*(2*n))*(2*n + 1)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{2 n \left(2 n - 1\right) \left(2 n + 1\right)}$$
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
$$a_{n} = \frac{1}{2 n \left(2 n - 1\right) \left(2 n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(2 n + 3\right) \left|{\frac{1}{2 n - 1}}\right|}{n}\right)$$
Let's take the limit
we find
$$\frac{1}{2 n \left(2 n - 1\right) \left(2 n + 1\right)}$$
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
$$a_{n} = \frac{1}{2 n \left(2 n - 1\right) \left(2 n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(2 n + 3\right) \left|{\frac{1}{2 n - 1}}\right|}{n}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
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