Mister Exam
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How to use it?
- Sum of series:
-
1/n*(n+1)(n+2)
- (-p+5h-2,5g)*1,5n
-
x²
- (xy)n
-
Identical expressions
- one /n*(n+ one)(n+ two)
- 1 divide by n multiply by (n plus 1)(n plus 2)
- one divide by n multiply by (n plus one)(n plus two)
- 1/n(n+1)(n+2)
- 1/nn+1n+2
- 1 divide by n*(n+1)(n+2)
-
Similar expressions
- 1/n*(n+1)(n-2)
- (n-1)/(n(n+1)(n+2))
- 1/(n*(n+1)*(n+2))
- 1/n*(n-1)(n+2)
- ((-1)^(n+1))/(n*(n+1)*(n+2))
- 1/(n*(n+1)*(n+2)*(n+3))
- 1/(n(n+1)(n+2))
Sum of series 1/n*(n+1)(n+2)
The solution
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oo ___ \ ` \ n + 1 ) -----*(n + 2) / n /__, n = 1
$$\sum_{n=1}^{\infty} \frac{n + 1}{n} \left(n + 2\right)$$
Sum(((n + 1)/n)*(n + 2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n + 1}{n} \left(n + 2\right)$$
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
$$a_{n} = \frac{\left(n + 1\right) \left(n + 2\right)}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 3\right)}\right)$$
Let's take the limit
we find
$$\frac{n + 1}{n} \left(n + 2\right)$$
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
$$a_{n} = \frac{\left(n + 1\right) \left(n + 2\right)}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 3\right)}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
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