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:
- x^n
-
1/n*(n+1)*(n+2)
-
1/((2n+5)(2n+7))
-
6/((3n-1)(3n+5))
- Graphing y =:
- 2/(x^2-2x)
-
Identical expressions
- two /(x^ two -2x)
- 2 divide by (x squared minus 2x)
- two divide by (x to the power of two minus 2x)
- 2/(x2-2x)
- 2/x2-2x
- 2/(x²-2x)
- 2/(x to the power of 2-2x)
- 2/x^2-2x
- 2 divide by (x^2-2x)
-
Similar expressions
- 2/(x^2+2x)
Sum of series 2/(x^2-2x)
The solution
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oo ____ \ ` \ 2 \ -------- / 2 / x - 2*x /___, n = 3
$$\sum_{n=3}^{\infty} \frac{2}{x^{2} - 2 x}$$
Sum(2/(x^2 - 2*x), (n, 3, oo))
The radius of convergence of the power series
Given number:
$$\frac{2}{x^{2} - 2 x}$$
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
$$a_{n} = \frac{2}{x^{2} - 2 x}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(2 \left|{\frac{\frac{x^{2}}{2} - x}{x^{2} - 2 x}}\right|\right)$$
Let's take the limit
we find
$$1 = 2 \left|{\frac{\frac{x^{2}}{2} - x}{x^{2} - 2 x}}\right|$$
$$1 = 2 \left|{\frac{\frac{x^{2}}{2} - x}{x^{2} - 2 x}}\right|$$
$$\frac{2}{x^{2} - 2 x}$$
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
$$a_{n} = \frac{2}{x^{2} - 2 x}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(2 \left|{\frac{\frac{x^{2}}{2} - x}{x^{2} - 2 x}}\right|\right)$$
Let's take the limit
we find
$$1 = 2 \left|{\frac{\frac{x^{2}}{2} - x}{x^{2} - 2 x}}\right|$$
$$1 = 2 \left|{\frac{\frac{x^{2}}{2} - x}{x^{2} - 2 x}}\right|$$
False
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