Mister Exam
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- 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:
-
(-1/2)^n
-
1/((2n+1)(2n+5))
-
1/(n+2)^4
-
1/(2n+5)(2n+7)
- Graphing y =:
- 1/(x*(x+1))
- Integral of d{x}:
-
1/(x*(x+1))
-
Identical expressions
- one /(x*(x+ one))
- 1 divide by (x multiply by (x plus 1))
- one divide by (x multiply by (x plus one))
- 1/(x(x+1))
- 1/xx+1
- 1 divide by (x*(x+1))
-
Similar expressions
- 1/(x*(x-1))
- 1/((x)(x+1)(x+2))
- 1/((x(x+1)))^(1/2)
- 1/(x*(x+1)*(x+2))
- 1/(x(x+1))
Sum of series 1/(x*(x+1))
The solution
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oo ___ \ ` \ 1 ) --------- / x*(x + 1) /__, x = 1
$$\sum_{x=1}^{\infty} \frac{1}{x \left(x + 1\right)}$$
Sum(1/(x*(x + 1)), (x, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{x \left(x + 1\right)}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \frac{1}{x \left(x + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{x + 2}{x}\right)$$
Let's take the limit
we find
$$\frac{1}{x \left(x + 1\right)}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \frac{1}{x \left(x + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{x + 2}{x}\right)$$
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