Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
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How to use it?
- Sum of series:
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1/((2n-1)(2n+3))
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(n+1)/(n^2*(n+2)^2)
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1/(nln^2n)
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log((n+1)/(n+2))
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Identical expressions
- (n+ one)/(n^ two *(n+ two)^ two)
- (n plus 1) divide by (n squared multiply by (n plus 2) squared )
- (n plus one) divide by (n to the power of two multiply by (n plus two) to the power of two)
- (n+1)/(n2*(n+2)2)
- n+1/n2*n+22
- (n+1)/(n²*(n+2)²)
- (n+1)/(n to the power of 2*(n+2) to the power of 2)
- (n+1)/(n^2(n+2)^2)
- (n+1)/(n2(n+2)2)
- n+1/n2n+22
- n+1/n^2n+2^2
- (n+1) divide by (n^2*(n+2)^2)
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Similar expressions
- (n-1)/(n^2*(n+2)^2)
- (n+1)/(n^2*(n-2)^2)
Sum of series (n+1)/(n^2*(n+2)^2)
The solution
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oo ____ \ ` \ n + 1 \ ----------- / 2 2 / n *(n + 2) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{n + 1}{n^{2} \left(n + 2\right)^{2}}$$
Sum((n + 1)/((n^2*(n + 2)^2)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n + 1}{n^{2} \left(n + 2\right)^{2}}$$
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
$$a_{n} = \frac{n + 1}{n^{2} \left(n + 2\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{3} \left(n + 3\right)^{2}}{n^{2} \left(n + 2\right)^{3}}\right)$$
Let's take the limit
we find
$$\frac{n + 1}{n^{2} \left(n + 2\right)^{2}}$$
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
$$a_{n} = \frac{n + 1}{n^{2} \left(n + 2\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{3} \left(n + 3\right)^{2}}{n^{2} \left(n + 2\right)^{3}}\right)$$
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