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:
-
(3/4)^n
-
1/((2n-1)*(2n+1))
-
1/(n*(n+3))
-
ln(n+1)/n
-
Identical expressions
- one /(n*(n- one))
- 1 divide by (n multiply by (n minus 1))
- one divide by (n multiply by (n minus one))
- 1/(n(n-1))
- 1/nn-1
- 1 divide by (n*(n-1))
-
Similar expressions
- 1/(n*(n-1)*(n-2))
- 1/n^(n-1)
- 1/((n^(n-1))+1)
- 1/(n*(n+1))
- 1/n*(n-1)
- 1/(n(n-1))*0.00405^2
Sum of series 1/(n*(n-1))
The solution
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oo ___ \ ` \ 1 ) --------- / n*(n - 1) /__, n = 2
$$\sum_{n=2}^{\infty} \frac{1}{n \left(n - 1\right)}$$
Sum(1/(n*(n - 1)), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{n \left(n - 1\right)}$$
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
$$a_{n} = \frac{1}{n \left(n - 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(n + 1\right) \left|{\frac{1}{n - 1}}\right|\right)$$
Let's take the limit
we find
$$\frac{1}{n \left(n - 1\right)}$$
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
$$a_{n} = \frac{1}{n \left(n - 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(n + 1\right) \left|{\frac{1}{n - 1}}\right|\right)$$
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