Mister Exam
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How to use it?
- Sum of series:
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arctg(1/n)/n!
-
arcsin1/sqrtn
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3*(pi/2)^(n-1)
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(1-cos(1/n))(√(n+1)-√n)
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Identical expressions
- three *(pi/ two)^(n- one)
- 3 multiply by ( Pi divide by 2) to the power of (n minus 1)
- three multiply by ( Pi divide by two) to the power of (n minus one)
- 3*(pi/2)(n-1)
- 3*pi/2n-1
- 3(pi/2)^(n-1)
- 3(pi/2)(n-1)
- 3pi/2n-1
- 3pi/2^n-1
- 3*(pi divide by 2)^(n-1)
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Similar expressions
- 3*(pi/2)^(n+1)
Sum of series 3*(pi/2)^(n-1)
The solution
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oo ____ \ ` \ n - 1 \ /pi\ / 3*|--| / \2 / /___, n = 1
$$\sum_{n=1}^{\infty} 3 \left(\frac{\pi}{2}\right)^{n - 1}$$
Sum(3*(pi/2)^(n - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$3 \left(\frac{\pi}{2}\right)^{n - 1}$$
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
$$a_{n} = 3 \left(\frac{\pi}{2}\right)^{n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{\pi}{2}\right)^{- n} \left(\frac{\pi}{2}\right)^{n - 1}\right)$$
Let's take the limit
we find
$$3 \left(\frac{\pi}{2}\right)^{n - 1}$$
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
$$a_{n} = 3 \left(\frac{\pi}{2}\right)^{n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{\pi}{2}\right)^{- n} \left(\frac{\pi}{2}\right)^{n - 1}\right)$$
Let's take the limit
we find
False
False
False
The rate of convergence of the power series
Numerical answer
The series diverges
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