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:
-
sqrt(2)/e^(2)
-
factorial(1000)/(factorial(450)*factorial(1000-450))
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atan(n)/n^2
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arctan1/n^2+n+1
-
Identical expressions
- arctan one /n^ two +n+1
- arc tangent of 1 divide by n squared plus n plus 1
- arc tangent of one divide by n to the power of two plus n plus 1
- arctan1/n2+n+1
- arctan1/n²+n+1
- arctan1/n to the power of 2+n+1
- arctan1 divide by n^2+n+1
-
Similar expressions
- arctan1/n^2-n+1
- arctan1/n^2+n-1
Sum of series arctan1/n^2+n+1
The solution
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oo ____ \ ` \ /atan(1) \ \ |------- + n + 1| / | 2 | / \ n / /___, n = 1
$$\sum_{n=1}^{\infty} \left(\left(n + \frac{\operatorname{atan}{\left(1 \right)}}{n^{2}}\right) + 1\right)$$
Sum(atan(1)/n^2 + n + 1, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(n + \frac{\operatorname{atan}{\left(1 \right)}}{n^{2}}\right) + 1$$
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
$$a_{n} = n + 1 + \frac{\pi}{4 n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n + 1 + \frac{\pi}{4 n^{2}}}{n + 2 + \frac{\pi}{4 \left(n + 1\right)^{2}}}\right)$$
Let's take the limit
we find
$$\left(n + \frac{\operatorname{atan}{\left(1 \right)}}{n^{2}}\right) + 1$$
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
$$a_{n} = n + 1 + \frac{\pi}{4 n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n + 1 + \frac{\pi}{4 n^{2}}}{n + 2 + \frac{\pi}{4 \left(n + 1\right)^{2}}}\right)$$
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