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