Mister Exam
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- Product of series
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How to use it?
- Sum of series:
-
n^2/3^n
-
1/((3*n+1)*(3*n+4))
- sin(n*x)/n^2
-
sin²n/√n³+5
-
Identical expressions
- n^ two / three ^n
- n squared divide by 3 to the power of n
- n to the power of two divide by three to the power of n
- n2/3n
- n²/3^n
- n to the power of 2/3 to the power of n
- n^2 divide by 3^n
Sum of series n^2/3^n
The solution
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oo ____ \ ` \ 2 \ n ) -- / n / 3 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{n^{2}}{3^{n}}$$
Sum(n^2/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n^{2}}{3^{n}}$$
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
$$a_{n} = n^{2}$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(\frac{n^{2}}{\left(n + 1\right)^{2}}\right)\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{n^{2}}{3^{n}}$$
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
$$a_{n} = n^{2}$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(\frac{n^{2}}{\left(n + 1\right)^{2}}\right)\right)$$
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