Mister Exam
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- Product of series
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How to use it?
- Sum of series:
-
1/2^n
-
((-3)^(n+1))/(2^(3n))
-
cos(n)
-
12(0.73)^(n-1)
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Identical expressions
- ((- three)^(n+ one))/(two ^(3n))
- (( minus 3) to the power of (n plus 1)) divide by (2 to the power of (3n))
- (( minus three) to the power of (n plus one)) divide by (two to the power of (3n))
- ((-3)(n+1))/(2(3n))
- -3n+1/23n
- -3^n+1/2^3n
- ((-3)^(n+1)) divide by (2^(3n))
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Similar expressions
- ((-3)^(n-1))/(2^(3n))
- ((3)^(n+1))/(2^(3n))
Sum of series ((-3)^(n+1))/(2^(3n))
The solution
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oo ____ \ ` \ n + 1 \ (-3) ) --------- / 3*n / 2 /___, n = 2
$$\sum_{n=2}^{\infty} \frac{\left(-3\right)^{n + 1}}{2^{3 n}}$$
Sum((-3)^(n + 1)/2^(3*n), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-3\right)^{n + 1}}{2^{3 n}}$$
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
$$a_{n} = \left(-3\right)^{n + 1}$$
and
$$x_{0} = -2$$
,
$$d = -3$$
,
$$c = 0$$
then
$$\frac{1}{R^{3}} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(3^{- n - 2} \cdot 3^{n + 1}\right)\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{\left(-3\right)^{n + 1}}{2^{3 n}}$$
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
$$a_{n} = \left(-3\right)^{n + 1}$$
and
$$x_{0} = -2$$
,
$$d = -3$$
,
$$c = 0$$
then
$$\frac{1}{R^{3}} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(3^{- n - 2} \cdot 3^{n + 1}\right)\right)$$
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