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:
- (a)^(i-1)(b)^(k-i)((a)^k+(b)^k)
-
sin(1/n)*(1-cos(1/n))
- sin(x/2^n)*cos(3x/2^n)
-
n^3/(lnn)^n
-
Identical expressions
- n^ three /(lnn)^n
- n cubed divide by (lnn) to the power of n
- n to the power of three divide by (lnn) to the power of n
- n3/(lnn)n
- n3/lnnn
- n³/(lnn)^n
- n to the power of 3/(lnn) to the power of n
- n^3/lnn^n
- n^3 divide by (lnn)^n
Sum of series n^3/(lnn)^n
The solution
You have entered
[src]
oo ____ \ ` \ 3 \ n ) ------- / n / log (n) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{n^{3}}{\log{\left(n \right)}^{n}}$$
Sum(n^3/log(n)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n^{3}}{\log{\left(n \right)}^{n}}$$
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
$$a_{n} = n^{3} \log{\left(n \right)}^{- n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n^{3} \log{\left(n + 1 \right)}^{n + 1}}{\left(n + 1\right)^{3} \left|{\log{\left(n \right)}^{n}}\right|}\right)$$
Let's take the limit
we find
$$\frac{n^{3}}{\log{\left(n \right)}^{n}}$$
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
$$a_{n} = n^{3} \log{\left(n \right)}^{- n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n^{3} \log{\left(n + 1 \right)}^{n + 1}}{\left(n + 1\right)^{3} \left|{\log{\left(n \right)}^{n}}\right|}\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ 3 -n / n *log (n) /__, n = 1
$$\sum_{n=1}^{\infty} n^{3} \log{\left(n \right)}^{- n}$$
Sum(n^3*log(n)^(-n), (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