Mister Exam
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How to use it?
- Sum of series:
- x^n/(1+x^n)
-
(-1)^n*(2*2)^(2*n)/factorial(2*n)
-
24/((5-3n)(2-3n))
-
arctan(n+3)-arctan(n+1)
-
Identical expressions
- one /(n(n+ three)(n+ six))
- 1 divide by (n(n plus 3)(n plus 6))
- one divide by (n(n plus three)(n plus six))
- 1/nn+3n+6
- 1 divide by (n(n+3)(n+6))
-
Similar expressions
- 1/(n(n-3)(n+6))
- 1/(n(n+3)(n-6))
Sum of series 1/(n(n+3)(n+6))
The solution
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[src]
oo ___ \ ` \ 1 ) ----------------- / n*(n + 3)*(n + 6) /__, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{n \left(n + 3\right) \left(n + 6\right)}$$
Sum(1/((n*(n + 3))*(n + 6)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{n \left(n + 3\right) \left(n + 6\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 3\right) \left(n + 6\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(n + 4\right) \left(n + 7\right)}{n \left(n + 3\right) \left(n + 6\right)}\right)$$
Let's take the limit
we find
$$\frac{1}{n \left(n + 3\right) \left(n + 6\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 3\right) \left(n + 6\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(n + 4\right) \left(n + 7\right)}{n \left(n + 3\right) \left(n + 6\right)}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
0 0 -10 + 40*e -40 + 22*e ---------------- + ---------------- / 0\ / 0\ 20*\-90 + 90*e / 24*\-45 + 45*e /
$$\frac{-40 + 22 e^{0}}{24 \left(-45 + 45 e^{0}\right)} + \frac{-10 + 40 e^{0}}{20 \left(-90 + 90 e^{0}\right)}$$
(-10 + 40*exp_polar(0))/(20*(-90 + 90*exp_polar(0))) + (-40 + 22*exp_polar(0))/(24*(-45 + 45*exp_polar(0)))
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