Mister Exam
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How to use it?
- Sum of series:
-
sqrt((n+4)/((n^4)+4))
- cos(kx)
- nx^n/(8n-9)*3^n
- ((-1)^(n-1))*(5^n+2)*x^n
-
Identical expressions
- one /(n+ one)/ln(n+ one)^ two
- 1 divide by (n plus 1) divide by ln(n plus 1) squared
- one divide by (n plus one) divide by ln(n plus one) to the power of two
- 1/(n+1)/ln(n+1)2
- 1/n+1/lnn+12
- 1/(n+1)/ln(n+1)²
- 1/(n+1)/ln(n+1) to the power of 2
- 1/n+1/lnn+1^2
- 1 divide by (n+1) divide by ln(n+1)^2
-
Similar expressions
- 1/(n+1)/ln(n-1)^2
- 1/(n-1)/ln(n+1)^2
Sum of series 1/(n+1)/ln(n+1)^2
The solution
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oo ____ \ ` \ 1 \ ------------------- / 2 / (n + 1)*log (n + 1) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}$$
Sum(1/((n + 1)*log(n + 1)^2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}$$
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
$$a_{n} = \frac{1}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right)$$
Let's take the limit
we find
$$\frac{1}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}$$
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
$$a_{n} = \frac{1}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 2\right) \log{\left(n + 2 \right)}^{2}}{\left(n + 1\right) \log{\left(n + 1 \right)}^{2}}\right)$$
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