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:
-
12/(n^2+4*n+3)
-
1/sqrt(n+1)
- x(1-x)^n
-
sqrt(3n+4)-sqrt(3n+1)
-
Identical expressions
- (arctg(one /sqrt(n- one)))/(n- one)
- (arctg(1 divide by square root of (n minus 1))) divide by (n minus 1)
- (arctg(one divide by square root of (n minus one))) divide by (n minus one)
- (arctg(1/√(n-1)))/(n-1)
- arctg1/sqrtn-1/n-1
- (arctg(1 divide by sqrt(n-1))) divide by (n-1)
-
Similar expressions
- (arctg(1/sqrt(n+1)))/(n-1)
- (arctg(1/sqrt(n-1)))/(n+1)
Sum of series (arctg(1/sqrt(n-1)))/(n-1)
The solution
You have entered
[src]
oo _____ \ ` \ / 1 \ \ atan|---------| \ | _______| / \\/ n - 1 / / --------------- / n - 1 /____, n = 1
$$\sum_{n=1}^{\infty} \frac{\operatorname{atan}{\left(\frac{1}{\sqrt{n - 1}} \right)}}{n - 1}$$
Sum(atan(1/(sqrt(n - 1)))/(n - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\operatorname{atan}{\left(\frac{1}{\sqrt{n - 1}} \right)}}{n - 1}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(\frac{1}{\sqrt{n - 1}} \right)}}{n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n \left|{\frac{\operatorname{atan}{\left(\frac{1}{\sqrt{n - 1}} \right)}}{n - 1}}\right|}{\operatorname{atan}{\left(\frac{1}{\sqrt{n}} \right)}}\right)$$
Let's take the limit
we find
$$\frac{\operatorname{atan}{\left(\frac{1}{\sqrt{n - 1}} \right)}}{n - 1}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(\frac{1}{\sqrt{n - 1}} \right)}}{n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n \left|{\frac{\operatorname{atan}{\left(\frac{1}{\sqrt{n - 1}} \right)}}{n - 1}}\right|}{\operatorname{atan}{\left(\frac{1}{\sqrt{n}} \right)}}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
oo _____ \ ` \ / 1 \ \ atan|----------| \ | ________| / \\/ -1 + n / / ---------------- / -1 + n /____, n = 1
$$\sum_{n=1}^{\infty} \frac{\operatorname{atan}{\left(\frac{1}{\sqrt{n - 1}} \right)}}{n - 1}$$
Sum(atan(1/sqrt(-1 + n))/(-1 + n), (n, 1, oo))
Numerical answer
[src]
Sum(atan(1/(sqrt(n - 1)))/(n - 1), (n, 1, oo))
Sum(atan(1/(sqrt(n - 1)))/(n - 1), (n, 1, oo))
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