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:
-
((-1)^(n+1))/n
-
sin(n)
-
(-1)^n*sqrt(n)/(n+100)
-
(0.15/0.05)^1.3
-
Identical expressions
- (sqrtn)arcsin(n+ one)/(n^ three - two)
- ( square root of n)arc sinus of (n plus 1) divide by (n cubed minus 2)
- ( square root of n)arc sinus of (n plus one) divide by (n to the power of three minus two)
- (√n)arcsin(n+1)/(n^3-2)
- (sqrtn)arcsin(n+1)/(n3-2)
- sqrtnarcsinn+1/n3-2
- (sqrtn)arcsin(n+1)/(n³-2)
- (sqrtn)arcsin(n+1)/(n to the power of 3-2)
- sqrtnarcsinn+1/n^3-2
- (sqrtn)arcsin(n+1) divide by (n^3-2)
-
Similar expressions
- (sqrtn)arcsin(n-1)/(n^3-2)
- (sqrtn)arcsin(n+1)/(n^3+2)
Sum of series (sqrtn)arcsin(n+1)/(n^3-2)
The solution
You have entered
[src]
oo ____ \ ` \ ___ \ \/ n *asin(n + 1) ) ----------------- / 3 / n - 2 /___, n = 2
$$\sum_{n=2}^{\infty} \frac{\sqrt{n} \operatorname{asin}{\left(n + 1 \right)}}{n^{3} - 2}$$
Sum((sqrt(n)*asin(n + 1))/(n^3 - 2), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sqrt{n} \operatorname{asin}{\left(n + 1 \right)}}{n^{3} - 2}$$
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
$$a_{n} = \frac{\sqrt{n} \operatorname{asin}{\left(n + 1 \right)}}{n^{3} - 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n} \left|{\frac{\left(\left(n + 1\right)^{3} - 2\right) \operatorname{asin}{\left(n + 1 \right)}}{\left(n^{3} - 2\right) \operatorname{asin}{\left(n + 2 \right)}}}\right|}{\sqrt{n + 1}}\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n} \left|{\frac{\left(\left(n + 1\right)^{3} - 2\right) \operatorname{asin}{\left(n + 1 \right)}}{\left(n^{3} - 2\right) \operatorname{asin}{\left(n + 2 \right)}}}\right|}{\sqrt{n + 1}}\right)$$
$$\frac{\sqrt{n} \operatorname{asin}{\left(n + 1 \right)}}{n^{3} - 2}$$
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
$$a_{n} = \frac{\sqrt{n} \operatorname{asin}{\left(n + 1 \right)}}{n^{3} - 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n} \left|{\frac{\left(\left(n + 1\right)^{3} - 2\right) \operatorname{asin}{\left(n + 1 \right)}}{\left(n^{3} - 2\right) \operatorname{asin}{\left(n + 2 \right)}}}\right|}{\sqrt{n + 1}}\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n} \left|{\frac{\left(\left(n + 1\right)^{3} - 2\right) \operatorname{asin}{\left(n + 1 \right)}}{\left(n^{3} - 2\right) \operatorname{asin}{\left(n + 2 \right)}}}\right|}{\sqrt{n + 1}}\right)$$
False
The answer
[src]
oo ____ \ ` \ ___ \ \/ n *asin(1 + n) ) ----------------- / 3 / -2 + n /___, n = 2
$$\sum_{n=2}^{\infty} \frac{\sqrt{n} \operatorname{asin}{\left(n + 1 \right)}}{n^{3} - 2}$$
Sum(sqrt(n)*asin(1 + n)/(-2 + n^3), (n, 2, oo))
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