Mister Exam
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- Product of series
-
How to use it?
- Sum of series:
- n^2*x^n
-
9/(9n^2+21n-8)
-
(n^3+n+5)/(n+6)
-
1/((5n-4)(5n+1))
-
Identical expressions
- nine /(9n^ two +21n- eight)
- 9 divide by (9n squared plus 21n minus 8)
- nine divide by (9n to the power of two plus 21n minus eight)
- 9/(9n2+21n-8)
- 9/9n2+21n-8
- 9/(9n²+21n-8)
- 9/(9n to the power of 2+21n-8)
- 9/9n^2+21n-8
- 9 divide by (9n^2+21n-8)
-
Similar expressions
- 9/(9n^2-21n-8)
- 9/(9n^2+21n+8)
Sum of series 9/(9n^2+21n-8)
The solution
You have entered
[src]
oo ____ \ ` \ 9 \ --------------- / 2 / 9*n + 21*n - 8 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{9}{\left(9 n^{2} + 21 n\right) - 8}$$
Sum(9/(9*n^2 + 21*n - 8), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{9}{\left(9 n^{2} + 21 n\right) - 8}$$
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
$$a_{n} = \frac{9}{9 n^{2} + 21 n - 8}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(21 n + 9 \left(n + 1\right)^{2} + 13\right) \left|{\frac{1}{9 n^{2} + 21 n - 8}}\right|\right)$$
Let's take the limit
we find
$$\frac{9}{\left(9 n^{2} + 21 n\right) - 8}$$
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
$$a_{n} = \frac{9}{9 n^{2} + 21 n - 8}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(21 n + 9 \left(n + 1\right)^{2} + 13\right) \left|{\frac{1}{9 n^{2} + 21 n - 8}}\right|\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
/ 0\ / 0\ 3*\-18 + 9*e /*Gamma(14/3) 9*\-1 + 4*e /*Gamma(14/3) ---------------------------- + ---------------------------- / 0\ / 0\ 22*\-20 + 20*e /*Gamma(11/3) 44*\-10 + 10*e /*Gamma(11/3)
$$\frac{3 \left(-18 + 9 e^{0}\right) \Gamma\left(\frac{14}{3}\right)}{22 \left(-20 + 20 e^{0}\right) \Gamma\left(\frac{11}{3}\right)} + \frac{9 \left(-1 + 4 e^{0}\right) \Gamma\left(\frac{14}{3}\right)}{44 \left(-10 + 10 e^{0}\right) \Gamma\left(\frac{11}{3}\right)}$$
3*(-18 + 9*exp_polar(0))*gamma(14/3)/(22*(-20 + 20*exp_polar(0))*gamma(11/3)) + 9*(-1 + 4*exp_polar(0))*gamma(14/3)/(44*(-10 + 10*exp_polar(0))*gamma(11/3))
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