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:
-
(1+(-3)^n)/6^n
- sin(kx)
-
nlnn/((n^2)-3)
-
1/(nln(n)ln(ln(n)))
-
Identical expressions
- nlnn/((n^ two)- three)
- nlnn divide by ((n squared ) minus 3)
- nlnn divide by ((n to the power of two) minus three)
- nlnn/((n2)-3)
- nlnn/n2-3
- nlnn/((n²)-3)
- nlnn/((n to the power of 2)-3)
- nlnn/n^2-3
- nlnn divide by ((n^2)-3)
-
Similar expressions
- nlnn/((n^2)+3)
Sum of series nlnn/((n^2)-3)
The solution
You have entered
[src]
oo ____ \ ` \ n*log(n) \ -------- / 2 / n - 3 /___, n = 2
$$\sum_{n=2}^{\infty} \frac{n \log{\left(n \right)}}{n^{2} - 3}$$
Sum((n*log(n))/(n^2 - 3), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{n \log{\left(n \right)}}{n^{2} - 3}$$
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
$$a_{n} = \frac{n \log{\left(n \right)}}{n^{2} - 3}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n \left|{\frac{\left(\left(n + 1\right)^{2} - 3\right) \log{\left(n \right)}}{n^{2} - 3}}\right|}{\left(n + 1\right) \log{\left(n + 1 \right)}}\right)$$
Let's take the limit
we find
$$\frac{n \log{\left(n \right)}}{n^{2} - 3}$$
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
$$a_{n} = \frac{n \log{\left(n \right)}}{n^{2} - 3}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n \left|{\frac{\left(\left(n + 1\right)^{2} - 3\right) \log{\left(n \right)}}{n^{2} - 3}}\right|}{\left(n + 1\right) \log{\left(n + 1 \right)}}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
oo ____ \ ` \ n*log(n) \ -------- / 2 / -3 + n /___, n = 2
$$\sum_{n=2}^{\infty} \frac{n \log{\left(n \right)}}{n^{2} - 3}$$
Sum(n*log(n)/(-3 + n^2), (n, 2, oo))
Numerical answer
The series diverges
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