Mister Exam
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- Product of series
-
How to use it?
- Sum of series:
-
9/(9n^2+3n-20)
-
0.5^n
-
56724.3*10^-1
- 1/(sqrt(k^4+7k^2+20))
-
Identical expressions
- nine /(9n^ two +3n- twenty)
- 9 divide by (9n squared plus 3n minus 20)
- nine divide by (9n to the power of two plus 3n minus twenty)
- 9/(9n2+3n-20)
- 9/9n2+3n-20
- 9/(9n²+3n-20)
- 9/(9n to the power of 2+3n-20)
- 9/9n^2+3n-20
- 9 divide by (9n^2+3n-20)
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Similar expressions
- 9/(9n^2+3n+20)
- 9/(9n^2-3n-20)
Sum of series 9/(9n^2+3n-20)
The solution
You have entered
[src]
oo ____ \ ` \ 9 \ --------------- / 2 / 9*n + 3*n - 20 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{9}{\left(9 n^{2} + 3 n\right) - 20}$$
Sum(9/(9*n^2 + 3*n - 20), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{9}{\left(9 n^{2} + 3 n\right) - 20}$$
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
$$a_{n} = \frac{9}{9 n^{2} + 3 n - 20}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(9 \left|{\frac{\frac{n}{3} + \left(n + 1\right)^{2} - \frac{17}{9}}{9 n^{2} + 3 n - 20}}\right|\right)$$
Let's take the limit
we find
$$\frac{9}{\left(9 n^{2} + 3 n\right) - 20}$$
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
$$a_{n} = \frac{9}{9 n^{2} + 3 n - 20}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(9 \left|{\frac{\frac{n}{3} + \left(n + 1\right)^{2} - \frac{17}{9}}{9 n^{2} + 3 n - 20}}\right|\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
/ 0\ / 0\
9*\-1 - 5*e /*Gamma(11/3) 3*\-9 + 18*e /*Gamma(11/3)
- --------------------------- - --------------------------
/ 0\ / 0\
16*\-10 + 10*e /*Gamma(8/3) 8*\-10 + 10*e /*Gamma(8/3)
$$- \frac{3 \left(-9 + 18 e^{0}\right) \Gamma\left(\frac{11}{3}\right)}{8 \left(-10 + 10 e^{0}\right) \Gamma\left(\frac{8}{3}\right)} - \frac{9 \left(- 5 e^{0} - 1\right) \Gamma\left(\frac{11}{3}\right)}{16 \left(-10 + 10 e^{0}\right) \Gamma\left(\frac{8}{3}\right)}$$
-9*(-1 - 5*exp_polar(0))*gamma(11/3)/(16*(-10 + 10*exp_polar(0))*gamma(8/3)) - 3*(-9 + 18*exp_polar(0))*gamma(11/3)/(8*(-10 + 10*exp_polar(0))*gamma(8/3))
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