Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
n^(1/n)
-
5^n
-
2(1/(n^2+5n+6))
- ((-1)^n)*((x^n)/(2n*n!))
-
Identical expressions
- two (one /(n^ two +5n+ six))
- 2(1 divide by (n squared plus 5n plus 6))
- two (one divide by (n to the power of two plus 5n plus six))
- 2(1/(n2+5n+6))
- 21/n2+5n+6
- 2(1/(n²+5n+6))
- 2(1/(n to the power of 2+5n+6))
- 21/n^2+5n+6
- 2(1 divide by (n^2+5n+6))
-
Similar expressions
- 2(1/(n^2-5n+6))
- 2(1/(n^2+5n-6))
Sum of series 2(1/(n^2+5n+6))
The solution
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oo ____ \ ` \ 2 \ ------------ / 2 / n + 5*n + 6 /___, n = 0
$$\sum_{n=0}^{\infty} \frac{2}{\left(n^{2} + 5 n\right) + 6}$$
Sum(2/(n^2 + 5*n + 6), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{2}{\left(n^{2} + 5 n\right) + 6}$$
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
$$a_{n} = \frac{2}{n^{2} + 5 n + 6}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{2 \left(\frac{5 n}{2} + \frac{\left(n + 1\right)^{2}}{2} + \frac{11}{2}\right)}{n^{2} + 5 n + 6}\right)$$
Let's take the limit
we find
$$\frac{2}{\left(n^{2} + 5 n\right) + 6}$$
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
$$a_{n} = \frac{2}{n^{2} + 5 n + 6}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{2 \left(\frac{5 n}{2} + \frac{\left(n + 1\right)^{2}}{2} + \frac{11}{2}\right)}{n^{2} + 5 n + 6}\right)$$
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