Mister Exam
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- Product of series
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How to use it?
- Sum of series:
- n^2*x^n
-
9/(9n^2+21n-8)
-
(n^3+n+5)/(n+6)
-
1/((5n-4)(5n+1))
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Identical expressions
- one /(2n- one)*(2n+ one)*(2n+ three)
- 1 divide by (2n minus 1) multiply by (2n plus 1) multiply by (2n plus 3)
- one divide by (2n minus one) multiply by (2n plus one) multiply by (2n plus three)
- 1/(2n-1)(2n+1)(2n+3)
- 1/2n-12n+12n+3
- 1 divide by (2n-1)*(2n+1)*(2n+3)
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Similar expressions
- 1/(2n-1)*(2n+1)*(2n-3)
- 1/(2n+1)*(2n+1)*(2n+3)
- 1/((2n-1)*(2n+1)*(2n+3))
- 1/(2n-1)*(2n-1)*(2n+3)
- 1/(2n-1)(2n+1)(2n+3)
Sum of series 1/(2n-1)*(2n+1)*(2n+3)
The solution
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oo ___ \ ` \ 2*n + 1 ) -------*(2*n + 3) / 2*n - 1 /__, n = 1
$$\sum_{n=1}^{\infty} \frac{2 n + 1}{2 n - 1} \left(2 n + 3\right)$$
Sum(((2*n + 1)/(2*n - 1))*(2*n + 3), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2 n + 1}{2 n - 1} \left(2 n + 3\right)$$
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
$$a_{n} = \frac{\left(2 n + 1\right) \left(2 n + 3\right)}{2 n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 1\right)^{2} \left|{\frac{1}{2 n - 1}}\right|}{2 n + 5}\right)$$
Let's take the limit
we find
$$\frac{2 n + 1}{2 n - 1} \left(2 n + 3\right)$$
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
$$a_{n} = \frac{\left(2 n + 1\right) \left(2 n + 3\right)}{2 n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 1\right)^{2} \left|{\frac{1}{2 n - 1}}\right|}{2 n + 5}\right)$$
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