Mister Exam
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How to use it?
- Sum of series:
- cos(xi+yi)
-
arctg(1/(n^2+n+1))
-
1+1/factorial(n)
-
(n+1)^n/(n^(n+2))
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Identical expressions
- arctg(one /(n^ two +n+ one))
- arctg(1 divide by (n squared plus n plus 1))
- arctg(one divide by (n to the power of two plus n plus one))
- arctg(1/(n2+n+1))
- arctg1/n2+n+1
- arctg(1/(n²+n+1))
- arctg(1/(n to the power of 2+n+1))
- arctg1/n^2+n+1
- arctg(1 divide by (n^2+n+1))
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Similar expressions
- arctg(1/(n^2-n+1))
- arctg(1/(n^2+n-1))
Sum of series arctg(1/(n^2+n+1))
The solution
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oo ____ \ ` \ / 1 \ \ atan|----------| / | 2 | / \n + n + 1/ /___, n = 1
$$\sum_{n=1}^{\infty} \operatorname{atan}{\left(\frac{1}{\left(n^{2} + n\right) + 1} \right)}$$
Sum(atan(1/(n^2 + n + 1)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\operatorname{atan}{\left(\frac{1}{\left(n^{2} + n\right) + 1} \right)}$$
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
$$a_{n} = \operatorname{atan}{\left(\frac{1}{n^{2} + n + 1} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{n^{2} + n + 1} \right)}}{\operatorname{atan}{\left(\frac{1}{n + \left(n + 1\right)^{2} + 2} \right)}}\right)$$
Let's take the limit
we find
$$\operatorname{atan}{\left(\frac{1}{\left(n^{2} + n\right) + 1} \right)}$$
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
$$a_{n} = \operatorname{atan}{\left(\frac{1}{n^{2} + n + 1} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{n^{2} + n + 1} \right)}}{\operatorname{atan}{\left(\frac{1}{n + \left(n + 1\right)^{2} + 2} \right)}}\right)$$
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