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*ln(n))
-
sin((n^2+3)/(n^3+2))^2
-
ln(1/n+2)
-
log((k+1)/(k+2))
-
Identical expressions
- sqrt(n+ one)/(n^ two +ln^2n)
- square root of (n plus 1) divide by (n squared plus ln squared n)
- square root of (n plus one) divide by (n to the power of two plus ln squared n)
- √(n+1)/(n^2+ln^2n)
- sqrt(n+1)/(n2+ln2n)
- sqrtn+1/n2+ln2n
- sqrt(n+1)/(n²+ln²n)
- sqrt(n+1)/(n to the power of 2+ln to the power of 2n)
- sqrtn+1/n^2+ln^2n
- sqrt(n+1) divide by (n^2+ln^2n)
-
Similar expressions
- sqrt(n-1)/(n^2+ln^2n)
- sqrt(n+1)/(n^2-ln^2n)
Sum of series sqrt(n+1)/(n^2+ln^2n)
The solution
You have entered
[src]
oo ____ \ ` \ _______ \ \/ n + 1 ) ------------ / 2 2 / n + log (n) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\sqrt{n + 1}}{n^{2} + \log{\left(n \right)}^{2}}$$
Sum(sqrt(n + 1)/(n^2 + log(n)^2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sqrt{n + 1}}{n^{2} + \log{\left(n \right)}^{2}}$$
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
$$a_{n} = \frac{\sqrt{n + 1}}{n^{2} + \log{\left(n \right)}^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n + 1} \left(\left(n + 1\right)^{2} + \log{\left(n + 1 \right)}^{2}\right)}{\sqrt{n + 2} \left(n^{2} + \log{\left(n \right)}^{2}\right)}\right)$$
Let's take the limit
we find
$$\frac{\sqrt{n + 1}}{n^{2} + \log{\left(n \right)}^{2}}$$
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
$$a_{n} = \frac{\sqrt{n + 1}}{n^{2} + \log{\left(n \right)}^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n + 1} \left(\left(n + 1\right)^{2} + \log{\left(n + 1 \right)}^{2}\right)}{\sqrt{n + 2} \left(n^{2} + \log{\left(n \right)}^{2}\right)}\right)$$
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