Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
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How to use it?
- Sum of series:
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200(0.98)^n-1
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(3x^2-2x+3)(x+1)^x
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arcctg((n+1)/(n^2-3))
- sin(x)/n
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Identical expressions
- arcctg((n+ one)/(n^ two - three))
- arcctg((n plus 1) divide by (n squared minus 3))
- arcctg((n plus one) divide by (n to the power of two minus three))
- arcctg((n+1)/(n2-3))
- arcctgn+1/n2-3
- arcctg((n+1)/(n²-3))
- arcctg((n+1)/(n to the power of 2-3))
- arcctgn+1/n^2-3
- arcctg((n+1) divide by (n^2-3))
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Similar expressions
- arcctg((n+1)/(n^2+3))
- arcctg((n-1)/(n^2-3))
Sum of series arcctg((n+1)/(n^2-3))
The solution
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oo ____ \ ` \ /n + 1 \ \ acot|------| / | 2 | / \n - 3/ /___, n = 1
$$\sum_{n=1}^{\infty} \operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}$$
Sum(acot((n + 1)/(n^2 - 3)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}$$
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
$$a_{n} = \operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}}{\operatorname{acot}{\left(\frac{n + 2}{\left(n + 1\right)^{2} - 3} \right)}}}\right|$$
Let's take the limit
we find
$$\operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}$$
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
$$a_{n} = \operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}}{\operatorname{acot}{\left(\frac{n + 2}{\left(n + 1\right)^{2} - 3} \right)}}}\right|$$
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