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/((3n-2)(3n+1))
-
(1000^n)/n!
- ((-1)^n*(3*cos(n*x)-pi*n*sin(n*x))-3*cos(n*x+pi*n/3))/n^2
-
(-1)^n*2^n/factorial(n)
-
Identical expressions
- one /(n^ two -3n+ two)
- 1 divide by (n squared minus 3n plus 2)
- one divide by (n to the power of two minus 3n plus two)
- 1/(n2-3n+2)
- 1/n2-3n+2
- 1/(n²-3n+2)
- 1/(n to the power of 2-3n+2)
- 1/n^2-3n+2
- 1 divide by (n^2-3n+2)
-
Similar expressions
- 1/(n^2+3n+2)
- 1/(n^2-3n-2)
Sum of series 1/(n^2-3n+2)
The solution
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oo ____ \ ` \ 1 \ ------------ / 2 / n - 3*n + 2 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(n^{2} - 3 n\right) + 2}$$
Sum(1/(n^2 - 3*n + 2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(n^{2} - 3 n\right) + 2}$$
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
$$a_{n} = \frac{1}{n^{2} - 3 n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{3 n - \left(n + 1\right)^{2} + 1}{n^{2} - 3 n + 2}}\right|$$
Let's take the limit
we find
$$\frac{1}{\left(n^{2} - 3 n\right) + 2}$$
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
$$a_{n} = \frac{1}{n^{2} - 3 n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{3 n - \left(n + 1\right)^{2} + 1}{n^{2} - 3 n + 2}}\right|$$
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