Mister Exam
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- Product of series
-
How to use it?
- Sum of series:
- ((-1)^n)*((x^n)/(2n*n!))
-
1/((n+6)*(n+7))
-
0.1
-
((n^2+5n-3)/(4n+7))^n
-
Identical expressions
- twenty-four /(9n^ two -12n+ five)
- 24 divide by (9n squared minus 12n plus 5)
- twenty minus four divide by (9n to the power of two minus 12n plus five)
- 24/(9n2-12n+5)
- 24/9n2-12n+5
- 24/(9n²-12n+5)
- 24/(9n to the power of 2-12n+5)
- 24/9n^2-12n+5
- 24 divide by (9n^2-12n+5)
-
Similar expressions
- 24/(9n^2+12n+5)
- 24/(9n^2-12n-5)
Sum of series 24/(9n^2-12n+5)
The solution
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oo ____ \ ` \ 24 \ --------------- / 2 / 9*n - 12*n + 5 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{24}{\left(9 n^{2} - 12 n\right) + 5}$$
Sum(24/(9*n^2 - 12*n + 5), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{24}{\left(9 n^{2} - 12 n\right) + 5}$$
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
$$a_{n} = \frac{24}{9 n^{2} - 12 n + 5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(- 12 n + 9 \left(n + 1\right)^{2} - 7\right) \left|{\frac{1}{9 n^{2} - 12 n + 5}}\right|\right)$$
Let's take the limit
we find
$$\frac{24}{\left(9 n^{2} - 12 n\right) + 5}$$
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
$$a_{n} = \frac{24}{9 n^{2} - 12 n + 5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(- 12 n + 9 \left(n + 1\right)^{2} - 7\right) \left|{\frac{1}{9 n^{2} - 12 n + 5}}\right|\right)$$
Let's take the limit
we find
True
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