Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
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How to use it?
- Sum of series:
- ((n^3+3n+1)^(1/2))arcsin(2/n^3)
-
3/((n+2)^2-1)
-
tan(pi/(4*n))
-
arctg(n)/(1+n^2)
-
Identical expressions
- arctg(n)/(one +n^ two)
- arctg(n) divide by (1 plus n squared )
- arctg(n) divide by (one plus n to the power of two)
- arctg(n)/(1+n2)
- arctgn/1+n2
- arctg(n)/(1+n²)
- arctg(n)/(1+n to the power of 2)
- arctgn/1+n^2
- arctg(n) divide by (1+n^2)
-
Similar expressions
- arctg(n)/(1-n^2)
Sum of series arctg(n)/(1+n^2)
The solution
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oo ____ \ ` \ atan(n) \ ------- / 2 / 1 + n /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\operatorname{atan}{\left(n \right)}}{n^{2} + 1}$$
Sum(atan(n)/(1 + n^2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\operatorname{atan}{\left(n \right)}}{n^{2} + 1}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(n \right)}}{n^{2} + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\left(n + 1\right)^{2} + 1\right) \operatorname{atan}{\left(n \right)}}{\left(n^{2} + 1\right) \operatorname{atan}{\left(n + 1 \right)}}\right)$$
Let's take the limit
we find
$$\frac{\operatorname{atan}{\left(n \right)}}{n^{2} + 1}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(n \right)}}{n^{2} + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\left(n + 1\right)^{2} + 1\right) \operatorname{atan}{\left(n \right)}}{\left(n^{2} + 1\right) \operatorname{atan}{\left(n + 1 \right)}}\right)$$
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