Mister Exam
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How to use it?
- Sum of series:
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38
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arctan(n)/(n!)^2
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arctg(5/n)/n!
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ln((n^2+n+3)/(n^2+n+2))
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Identical expressions
- arctg(five /n)/n!
- arctg(5 divide by n) divide by n!
- arctg(five divide by n) divide by n!
- arctg5/n/n!
Sum of series arctg(5/n)/n!
The solution
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oo ____ \ ` \ /5\ \ atan|-| ) \n/ / ------- / n! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\operatorname{atan}{\left(\frac{5}{n} \right)}}{n!}$$
Sum(atan(5/n)/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\operatorname{atan}{\left(\frac{5}{n} \right)}}{n!}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(\frac{5}{n} \right)}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{\left(n + 1\right)!}{n!}}\right| \operatorname{atan}{\left(\frac{5}{n} \right)}}{\operatorname{atan}{\left(\frac{5}{n + 1} \right)}}\right)$$
Let's take the limit
we find
$$\frac{\operatorname{atan}{\left(\frac{5}{n} \right)}}{n!}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(\frac{5}{n} \right)}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{\left(n + 1\right)!}{n!}}\right| \operatorname{atan}{\left(\frac{5}{n} \right)}}{\operatorname{atan}{\left(\frac{5}{n + 1} \right)}}\right)$$
Let's take the limit
we find
False
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