Mister Exam
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How to use it?
- Sum of series:
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n/2^n
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7^(n+1)/(4^(n-2)*9^n)
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1/nln(n)
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(-1)^(n+1)/n
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Identical expressions
- seven ^(n+ one)/(four ^(n- two)* nine ^n)
- 7 to the power of (n plus 1) divide by (4 to the power of (n minus 2) multiply by 9 to the power of n)
- seven to the power of (n plus one) divide by (four to the power of (n minus two) multiply by nine to the power of n)
- 7(n+1)/(4(n-2)*9n)
- 7n+1/4n-2*9n
- 7^(n+1)/(4^(n-2)9^n)
- 7(n+1)/(4(n-2)9n)
- 7n+1/4n-29n
- 7^n+1/4^n-29^n
- 7^(n+1) divide by (4^(n-2)*9^n)
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Similar expressions
- 7^(n+1)/(4^(n+2)*9^n)
- 7^(n-1)/(4^(n-2)*9^n)
Sum of series 7^(n+1)/(4^(n-2)*9^n)
The solution
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oo ____ \ ` \ n + 1 \ 7 ) --------- / n - 2 n / 4 *9 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{7^{n + 1}}{4^{n - 2} \cdot 9^{n}}$$
Sum(7^(n + 1)/((4^(n - 2)*9^n)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{7^{n + 1}}{4^{n - 2} \cdot 9^{n}}$$
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
$$a_{n} = 4^{2 - n} 7^{n + 1}$$
and
$$x_{0} = -9$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-9 + \lim_{n \to \infty}\left(4^{2 - n} 4^{n - 1} \cdot 7^{- n - 2} \cdot 7^{n + 1}\right)\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{7^{n + 1}}{4^{n - 2} \cdot 9^{n}}$$
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
$$a_{n} = 4^{2 - n} 7^{n + 1}$$
and
$$x_{0} = -9$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-9 + \lim_{n \to \infty}\left(4^{2 - n} 4^{n - 1} \cdot 7^{- n - 2} \cdot 7^{n + 1}\right)\right)$$
Let's take the limit
we find
False
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