Mister Exam
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- Product of series
-
How to use it?
- Sum of series:
-
1/(n*ln(n))
-
sin((n^2+3)/(n^3+2))^2
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1/(2*n+5)*(2*n+7)
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ln(1/n+2)
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Identical expressions
- one /(two *n+ five)*(two *n+ seven)
- 1 divide by (2 multiply by n plus 5) multiply by (2 multiply by n plus 7)
- one divide by (two multiply by n plus five) multiply by (two multiply by n plus seven)
- 1/(2n+5)(2n+7)
- 1/2n+52n+7
- 1 divide by (2*n+5)*(2*n+7)
-
Similar expressions
- 1/((2n+5)(2n+7))
- 1/(2n+5)*(2n+7)
- 1/((2n+5)*(2n+7))
- 1/(2*n+5)*(2*n-7)
- 1/(2n+5)(2n+7)
- 1/(2*n+5)(2*n+7)
- 1/(2*n-5)*(2*n+7)
Sum of series 1/(2*n+5)*(2*n+7)
The solution
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oo ___ \ ` \ 2*n + 7 ) ------- / 2*n + 5 /__, n = 0
$$\sum_{n=0}^{\infty} \frac{2 n + 7}{2 n + 5}$$
Sum((2*n + 7)/(2*n + 5), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{2 n + 7}{2 n + 5}$$
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
$$a_{n} = \frac{2 n + 7}{2 n + 5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 7\right)^{2}}{\left(2 n + 5\right) \left(2 n + 9\right)}\right)$$
Let's take the limit
we find
$$\frac{2 n + 7}{2 n + 5}$$
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
$$a_{n} = \frac{2 n + 7}{2 n + 5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 7\right)^{2}}{\left(2 n + 5\right) \left(2 n + 9\right)}\right)$$
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