Mister Exam
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- Product of series
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How to use it?
- Sum of series:
-
1/n^n
-
(2n+1)/(n(n+2)(n+4))
-
1/(n(n+2)(n+1))
-
((n^2+5n-3)/(4n+7))^n
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Identical expressions
- one /((n(n+ one)(n+ two)(n+ three)))
- 1 divide by ((n(n plus 1)(n plus 2)(n plus 3)))
- one divide by ((n(n plus one)(n plus two)(n plus three)))
- 1/nn+1n+2n+3
- 1 divide by ((n(n+1)(n+2)(n+3)))
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Similar expressions
- 1/((n(n+1)(n-2)(n+3)))
- 1/((n(n+1)(n+2)(n-3)))
- 1/((n(n-1)(n+2)(n+3)))
Sum of series 1/((n(n+1)(n+2)(n+3)))
The solution
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oo ___ \ ` \ 1 ) ------------------------- / n*(n + 1)*(n + 2)*(n + 3) /__, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}$$
Sum(1/(((n*(n + 1))*(n + 2))*(n + 3)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n + 4}{n}\right)$$
Let's take the limit
we find
$$\frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n + 4}{n}\right)$$
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