Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
-
How to use it?
- Sum of series:
- x^n/n
-
(5^(1/sqrtn)-1)^3
-
arctan(n+3)-arctan(n+1)
- sin(x)/n
-
Identical expressions
- arctan(n+ three)-arctan(n+ one)
- arc tangent of (n plus 3) minus arc tangent of (n plus 1)
- arc tangent of (n plus three) minus arc tangent of (n plus one)
- arctann+3-arctann+1
-
Similar expressions
- arctan(n+3)+arctan(n+1)
- arctan(n+3)-arctan(n-1)
- arctan(n-3)-arctan(n+1)
Sum of series arctan(n+3)-arctan(n+1)
The solution
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[src]
oo __ \ ` ) (atan(n + 3) - atan(n + 1)) /_, n = 1
$$\sum_{n=1}^{\infty} \left(- \operatorname{atan}{\left(n + 1 \right)} + \operatorname{atan}{\left(n + 3 \right)}\right)$$
Sum(atan(n + 3) - atan(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$- \operatorname{atan}{\left(n + 1 \right)} + \operatorname{atan}{\left(n + 3 \right)}$$
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
$$a_{n} = - \operatorname{atan}{\left(n + 1 \right)} + \operatorname{atan}{\left(n + 3 \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\operatorname{atan}{\left(n + 1 \right)} - \operatorname{atan}{\left(n + 3 \right)}}{\operatorname{atan}{\left(n + 2 \right)} - \operatorname{atan}{\left(n + 4 \right)}}}\right|$$
Let's take the limit
we find
$$- \operatorname{atan}{\left(n + 1 \right)} + \operatorname{atan}{\left(n + 3 \right)}$$
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
$$a_{n} = - \operatorname{atan}{\left(n + 1 \right)} + \operatorname{atan}{\left(n + 3 \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\operatorname{atan}{\left(n + 1 \right)} - \operatorname{atan}{\left(n + 3 \right)}}{\operatorname{atan}{\left(n + 2 \right)} - \operatorname{atan}{\left(n + 4 \right)}}}\right|$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
pi - atan(2) - atan(3)
$$- \operatorname{atan}{\left(3 \right)} - \operatorname{atan}{\left(2 \right)} + \pi$$
pi - atan(2) - atan(3)
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