Mister Exam
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How to use it?
- Sum of series:
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((-1)^(n+1))/n
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(1+1/n)^n
-
sin(sin(n)/n^(1/3))
-
3/(8^(n-1))
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Identical expressions
- (one)/(n^ two +13n+ forty)
- (1) divide by (n squared plus 13n plus 40)
- (one) divide by (n to the power of two plus 13n plus forty)
- (1)/(n2+13n+40)
- 1/n2+13n+40
- (1)/(n²+13n+40)
- (1)/(n to the power of 2+13n+40)
- 1/n^2+13n+40
- (1) divide by (n^2+13n+40)
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Similar expressions
- (1)/(n^2-13n+40)
- (1)/(n^2+13n-40)
Sum of series (1)/(n^2+13n+40)
The solution
You have entered
[src]
oo ____ \ ` \ 1 \ -------------- / 2 / n + 13*n + 40 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(n^{2} + 13 n\right) + 40}$$
Sum(1/(n^2 + 13*n + 40), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(n^{2} + 13 n\right) + 40}$$
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
$$a_{n} = \frac{1}{n^{2} + 13 n + 40}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{13 n + \left(n + 1\right)^{2} + 53}{n^{2} + 13 n + 40}\right)$$
Let's take the limit
we find
$$\frac{1}{\left(n^{2} + 13 n\right) + 40}$$
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
$$a_{n} = \frac{1}{n^{2} + 13 n + 40}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{13 n + \left(n + 1\right)^{2} + 53}{n^{2} + 13 n + 40}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
0 / 0\ -25 + 21*e 2*\-3 + 4*e / ---------------- + --------------- / 0\ / 0\ 18*\-28 + 28*e / 9*\-28 + 28*e /
$$\frac{-25 + 21 e^{0}}{18 \left(-28 + 28 e^{0}\right)} + \frac{2 \left(-3 + 4 e^{0}\right)}{9 \left(-28 + 28 e^{0}\right)}$$
(-25 + 21*exp_polar(0))/(18*(-28 + 28*exp_polar(0))) + 2*(-3 + 4*exp_polar(0))/(9*(-28 + 28*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