Mister Exam
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How to use it?
- Sum of series:
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(2n-1)/2^n
-
6/(9n^2+6n-8)
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sqrt((n+4)/((n^4)+4))
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1/(2*n-1)*(2*n+1)
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Identical expressions
- (two ^(n+ one))/(three ^(n- two)* seven ^n)
- (2 to the power of (n plus 1)) divide by (3 to the power of (n minus 2) multiply by 7 to the power of n)
- (two to the power of (n plus one)) divide by (three to the power of (n minus two) multiply by seven to the power of n)
- (2(n+1))/(3(n-2)*7n)
- 2n+1/3n-2*7n
- (2^(n+1))/(3^(n-2)7^n)
- (2(n+1))/(3(n-2)7n)
- 2n+1/3n-27n
- 2^n+1/3^n-27^n
- (2^(n+1)) divide by (3^(n-2)*7^n)
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Similar expressions
- (2^(n+1))/(3^(n+2)*7^n)
- (2^(n-1))/(3^(n-2)*7^n)
Sum of series (2^(n+1))/(3^(n-2)*7^n)
The solution
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oo ____ \ ` \ n + 1 \ 2 ) --------- / n - 2 n / 3 *7 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{2^{n + 1}}{3^{n - 2} \cdot 7^{n}}$$
Sum(2^(n + 1)/((3^(n - 2)*7^n)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2^{n + 1}}{3^{n - 2} \cdot 7^{n}}$$
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
$$a_{n} = 2^{n + 1} \cdot 3^{2 - n}$$
and
$$x_{0} = -7$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-7 + \lim_{n \to \infty}\left(2^{- n - 2} \cdot 2^{n + 1} \cdot 3^{2 - n} 3^{n - 1}\right)\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{2^{n + 1}}{3^{n - 2} \cdot 7^{n}}$$
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
$$a_{n} = 2^{n + 1} \cdot 3^{2 - n}$$
and
$$x_{0} = -7$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-7 + \lim_{n \to \infty}\left(2^{- n - 2} \cdot 2^{n + 1} \cdot 3^{2 - n} 3^{n - 1}\right)\right)$$
Let's take the limit
we find
False
$$R = 0$$
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