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^2*x^n
-
9/(9n^2+21n-8)
-
(n^3+n+5)/(n+6)
-
1/((5n-4)(5n+1))
-
Identical expressions
- (n^ three +n+ five)/(n+ six)
- (n cubed plus n plus 5) divide by (n plus 6)
- (n to the power of three plus n plus five) divide by (n plus six)
- (n3+n+5)/(n+6)
- n3+n+5/n+6
- (n³+n+5)/(n+6)
- (n to the power of 3+n+5)/(n+6)
- n^3+n+5/n+6
- (n^3+n+5) divide by (n+6)
-
Similar expressions
- (n^3+n-5)/(n+6)
- (n^3+n+5)/(n-6)
- (n^3-n+5)/(n+6)
Sum of series (n^3+n+5)/(n+6)
The solution
You have entered
[src]
oo ____ \ ` \ 3 \ n + n + 5 / ---------- / n + 6 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(n^{3} + n\right) + 5}{n + 6}$$
Sum((n^3 + n + 5)/(n + 6), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(n^{3} + n\right) + 5}{n + 6}$$
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
$$a_{n} = \frac{n^{3} + n + 5}{n + 6}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 7\right) \left(n^{3} + n + 5\right)}{\left(n + 6\right) \left(n + \left(n + 1\right)^{3} + 6\right)}\right)$$
Let's take the limit
we find
$$\frac{\left(n^{3} + n\right) + 5}{n + 6}$$
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
$$a_{n} = \frac{n^{3} + n + 5}{n + 6}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 7\right) \left(n^{3} + n + 5\right)}{\left(n + 6\right) \left(n + \left(n + 1\right)^{3} + 6\right)}\right)$$
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