Mister Exam
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- Product of series
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How to use it?
- Sum of series:
-
(3^n+4^n)/12^n
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(2n+1)/(n^2(n+1)^2)
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sin^2((n^2+3)/(n^3+2))
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ln(n)/n^1.01-1/n^1.01
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Identical expressions
- (two n+ one)/(n^ two (n+ one)^2)
- (2n plus 1) divide by (n squared (n plus 1) squared )
- (two n plus one) divide by (n to the power of two (n plus one) squared )
- (2n+1)/(n2(n+1)2)
- 2n+1/n2n+12
- (2n+1)/(n²(n+1)²)
- (2n+1)/(n to the power of 2(n+1) to the power of 2)
- 2n+1/n^2n+1^2
- (2n+1) divide by (n^2(n+1)^2)
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Similar expressions
- (2n-1)/(n^2(n+1)^2)
- (2n+1)/(n^2(n-1)^2)
Sum of series (2n+1)/(n^2(n+1)^2)
The solution
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oo ____ \ ` \ 2*n + 1 \ ----------- / 2 2 / n *(n + 1) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{2 n + 1}{n^{2} \left(n + 1\right)^{2}}$$
Sum((2*n + 1)/((n^2*(n + 1)^2)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2 n + 1}{n^{2} \left(n + 1\right)^{2}}$$
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
$$a_{n} = \frac{2 n + 1}{n^{2} \left(n + 1\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 2\right)^{2} \left(2 n + 1\right)}{n^{2} \left(2 n + 3\right)}\right)$$
Let's take the limit
we find
$$\frac{2 n + 1}{n^{2} \left(n + 1\right)^{2}}$$
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
$$a_{n} = \frac{2 n + 1}{n^{2} \left(n + 1\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 2\right)^{2} \left(2 n + 1\right)}{n^{2} \left(2 n + 3\right)}\right)$$
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