Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
exp(-0.01*n)
-
cos(pin)
-
cos(2*n)/(n*n^(1/2))
- arctg(1/x^(1/3))
-
Identical expressions
- atan(one -pi/ two)^n
- arc tangent of gent of (1 minus Pi divide by 2) to the power of n
- arc tangent of gent of (one minus Pi divide by two) to the power of n
- atan(1-pi/2)n
- atan1-pi/2n
- atan1-pi/2^n
- atan(1-pi divide by 2)^n
-
Similar expressions
- atan(1+pi/2)^n
- arctan(1-pi/2)^n
Sum of series atan(1-pi/2)^n
The solution
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[src]
oo ___ \ ` \ n/ pi\ ) atan |1 - --| / \ 2 / /__, n = 1
$$\sum_{n=1}^{\infty} \operatorname{atan}^{n}{\left(- \frac{\pi}{2} + 1 \right)}$$
Sum(atan(1 - pi/2)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\operatorname{atan}^{n}{\left(- \frac{\pi}{2} + 1 \right)}$$
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
$$a_{n} = 1$$
and
$$x_{0} = - \operatorname{atan}{\left(1 - \frac{\pi}{2} \right)}$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- \operatorname{atan}{\left(1 - \frac{\pi}{2} \right)} + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
$$\operatorname{atan}^{n}{\left(- \frac{\pi}{2} + 1 \right)}$$
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
$$a_{n} = 1$$
and
$$x_{0} = - \operatorname{atan}{\left(1 - \frac{\pi}{2} \right)}$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- \operatorname{atan}{\left(1 - \frac{\pi}{2} \right)} + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
False
The rate of convergence of the power series
The answer
[src]
/ pi\
atan|1 - --|
\ 2 /
----------------
/ pi\
1 - atan|1 - --|
\ 2 /
$$\frac{\operatorname{atan}{\left(1 - \frac{\pi}{2} \right)}}{1 - \operatorname{atan}{\left(1 - \frac{\pi}{2} \right)}}$$
atan(1 - pi/2)/(1 - atan(1 - pi/2))
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