Mister Exam
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How to use it?
- Sum of series:
-
(9835^(2k/9835))/(9835+9835^(2k/9835))
- x^2/k^3/2
- (x-4)^n/sqrt(n)
-
(2*3^(n-1)+8^(n+1))/12^n
-
Identical expressions
- one /(n*(lnn+ one)^ two)
- 1 divide by (n multiply by (lnn plus 1) squared )
- one divide by (n multiply by (lnn plus one) to the power of two)
- 1/(n*(lnn+1)2)
- 1/n*lnn+12
- 1/(n*(lnn+1)²)
- 1/(n*(lnn+1) to the power of 2)
- 1/(n(lnn+1)^2)
- 1/(n(lnn+1)2)
- 1/nlnn+12
- 1/nlnn+1^2
- 1 divide by (n*(lnn+1)^2)
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Similar expressions
- 1/(n*(lnn-1)^2)
Sum of series 1/(n*(lnn+1)^2)
The solution
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oo ____ \ ` \ 1 \ --------------- / 2 / n*(log(n) + 1) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{n \left(\log{\left(n \right)} + 1\right)^{2}}$$
Sum(1/(n*(log(n) + 1)^2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{n \left(\log{\left(n \right)} + 1\right)^{2}}$$
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
$$a_{n} = \frac{1}{n \left(\log{\left(n \right)} + 1\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(\log{\left(n + 1 \right)} + 1\right)^{2} \left|{\frac{1}{\left(\log{\left(n \right)} + 1\right)^{2}}}\right|}{n}\right)$$
Let's take the limit
we find
$$\frac{1}{n \left(\log{\left(n \right)} + 1\right)^{2}}$$
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
$$a_{n} = \frac{1}{n \left(\log{\left(n \right)} + 1\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(\log{\left(n + 1 \right)} + 1\right)^{2} \left|{\frac{1}{\left(\log{\left(n \right)} + 1\right)^{2}}}\right|}{n}\right)$$
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