Mister Exam
Other calculators
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- Product of series
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How to use it?
- Sum of series:
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1/(n(n+1))
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1/(25n^2+5n-6)
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(-1)^(n+1)/(n+1)!
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(3^(n+3))/(4^(n-1)*5^n)
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Identical expressions
- one /(two 5n^2+5n- six)
- 1 divide by (25n squared plus 5n minus 6)
- one divide by (two 5n squared plus 5n minus six)
- 1/(25n2+5n-6)
- 1/25n2+5n-6
- 1/(25n²+5n-6)
- 1/(25n to the power of 2+5n-6)
- 1/25n^2+5n-6
- 1 divide by (25n^2+5n-6)
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Similar expressions
- 1/(25n^2+5n+6)
- 1/(25n^2-5n-6)
Sum of series 1/(25n^2+5n-6)
The solution
You have entered
[src]
oo ____ \ ` \ 1 \ --------------- / 2 / 25*n + 5*n - 6 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(25 n^{2} + 5 n\right) - 6}$$
Sum(1/(25*n^2 + 5*n - 6), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(25 n^{2} + 5 n\right) - 6}$$
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
$$a_{n} = \frac{1}{25 n^{2} + 5 n - 6}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5 n + 25 \left(n + 1\right)^{2} - 1\right) \left|{\frac{1}{25 n^{2} + 5 n - 6}}\right|\right)$$
Let's take the limit
we find
$$\frac{1}{\left(25 n^{2} + 5 n\right) - 6}$$
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
$$a_{n} = \frac{1}{25 n^{2} + 5 n - 6}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5 n + 25 \left(n + 1\right)^{2} - 1\right) \left|{\frac{1}{25 n^{2} + 5 n - 6}}\right|\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
Gamma(13/5) ------------- 24*Gamma(8/5)
$$\frac{\Gamma\left(\frac{13}{5}\right)}{24 \Gamma\left(\frac{8}{5}\right)}$$
gamma(13/5)/(24*gamma(8/5))
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