Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
(4*9^n-2^n)/18^n
-
1/((3*n+1)*(3*n+4))
- sqrt(n)*(x+3)^n/n^2+1
-
ln((n(n+2))/(n+1)^2)
-
Identical expressions
- (- twelve /(9n- fifteen)+ four /(3n+ one))
- ( minus 12 divide by (9n minus 15) plus 4 divide by (3n plus 1))
- ( minus twelve divide by (9n minus fifteen) plus four divide by (3n plus one))
- -12/9n-15+4/3n+1
- (-12 divide by (9n-15)+4 divide by (3n+1))
-
Similar expressions
- (-12/(9n-15)+4/(3n-1))
- (-12/(9n-15)-4/(3n+1))
- (-12/(9n+15)+4/(3n+1))
- (12/(9n-15)+4/(3n+1))
Sum of series (-12/(9n-15)+4/(3n+1))
The solution
You have entered
[src]
oo ___ \ ` \ / 12 4 \ ) |- -------- + -------| / \ 9*n - 15 3*n + 1/ /__, n = 1
$$\sum_{n=1}^{\infty} \left(- \frac{12}{9 n - 15} + \frac{4}{3 n + 1}\right)$$
Sum(-12/(9*n - 15) + 4/(3*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$- \frac{12}{9 n - 15} + \frac{4}{3 n + 1}$$
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
$$a_{n} = - \frac{12}{9 n - 15} + \frac{4}{3 n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 4\right) \left|{9 n - 6}\right|}{\left(3 n + 1\right) \left|{9 n - 15}\right|}\right)$$
Let's take the limit
we find
$$- \frac{12}{9 n - 15} + \frac{4}{3 n + 1}$$
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
$$a_{n} = - \frac{12}{9 n - 15} + \frac{4}{3 n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 4\right) \left|{9 n - 6}\right|}{\left(3 n + 1\right) \left|{9 n - 15}\right|}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
-3*Gamma(7/3) ------------- 2*Gamma(4/3)
$$- \frac{3 \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{4}{3}\right)}$$
-3*gamma(7/3)/(2*gamma(4/3))
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