Mister Exam
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- Product of series
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How to use it?
- Sum of series:
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sin(3n)/(7n)^(1/5)
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arctg1/(2n^2)
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cos(i*n)/2^n
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(8/((2*n+1)*(2*n+1)*3.1416*3.1416))*exp(-(2*n+1)*0.197*3.1416*3.1416*0.25)
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Identical expressions
- pi^n*((n- three)/(four *n- two))^n
- Pi to the power of n multiply by ((n minus 3) divide by (4 multiply by n minus 2)) to the power of n
- Pi to the power of n multiply by ((n minus three) divide by (four multiply by n minus two)) to the power of n
- pin*((n-3)/(4*n-2))n
- pin*n-3/4*n-2n
- pi^n((n-3)/(4n-2))^n
- pin((n-3)/(4n-2))n
- pinn-3/4n-2n
- pi^nn-3/4n-2^n
- pi^n*((n-3) divide by (4*n-2))^n
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Similar expressions
- pi^n*((n-3)/(4*n+2))^n
- pi^n*((n+3)/(4*n-2))^n
Sum of series pi^n*((n-3)/(4*n-2))^n
The solution
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oo ____ \ ` \ n \ n / n - 3 \ / pi *|-------| / \4*n - 2/ /___, n = 1
$$\sum_{n=1}^{\infty} \pi^{n} \left(\frac{n - 3}{4 n - 2}\right)^{n}$$
Sum(pi^n*((n - 3)/(4*n - 2))^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\pi^{n} \left(\frac{n - 3}{4 n - 2}\right)^{n}$$
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
$$a_{n} = \left(\frac{n - 3}{4 n - 2}\right)^{n}$$
and
$$x_{0} = - \pi$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- \pi + \lim_{n \to \infty} \left|{\left(\frac{n - 3}{4 n - 2}\right)^{n} \left(\frac{n - 2}{4 n + 2}\right)^{- n - 1}}\right|\right)$$
Let's take the limit
we find
$$\pi^{n} \left(\frac{n - 3}{4 n - 2}\right)^{n}$$
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
$$a_{n} = \left(\frac{n - 3}{4 n - 2}\right)^{n}$$
and
$$x_{0} = - \pi$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- \pi + \lim_{n \to \infty} \left|{\left(\frac{n - 3}{4 n - 2}\right)^{n} \left(\frac{n - 2}{4 n + 2}\right)^{- n - 1}}\right|\right)$$
Let's take the limit
we find
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