Mister Exam
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- 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:
-
sqrt(1+1)/1
-
arctg(5/n)/n!
-
cos(pin)
-
arctg^n(1/n)
-
Identical expressions
- arctg^n(one /n)
- arctg to the power of n(1 divide by n)
- arctg to the power of n(one divide by n)
- arctgn(1/n)
- arctgn1/n
- arctg^n1/n
- arctg^n(1 divide by n)
Sum of series arctg^n(1/n)
The solution
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oo ___ \ ` \ n/1\ ) atan |-| / \n/ /__, n = 1
$$\sum_{n=1}^{\infty} \operatorname{atan}^{n}{\left(\frac{1}{n} \right)}$$
Sum(atan(1/n)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\operatorname{atan}^{n}{\left(\frac{1}{n} \right)}$$
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
$$a_{n} = \operatorname{atan}^{n}{\left(\frac{1}{n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\operatorname{atan}^{n}{\left(\frac{1}{n} \right)} \operatorname{atan}^{- n - 1}{\left(\frac{1}{n + 1} \right)}\right)$$
Let's take the limit
we find
$$\operatorname{atan}^{n}{\left(\frac{1}{n} \right)}$$
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
$$a_{n} = \operatorname{atan}^{n}{\left(\frac{1}{n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\operatorname{atan}^{n}{\left(\frac{1}{n} \right)} \operatorname{atan}^{- n - 1}{\left(\frac{1}{n + 1} \right)}\right)$$
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