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:
-
(2/9)^n
-
n^2/2^n
-
(4^(n+1)-10^n)/20^n
- sin(1/(n*sqrt(n)))*x^n
-
Identical expressions
- one /((ln(n+ two))^n)
- 1 divide by ((ln(n plus 2)) to the power of n)
- one divide by ((ln(n plus two)) to the power of n)
- 1/((ln(n+2))n)
- 1/lnn+2n
- 1/lnn+2^n
- 1 divide by ((ln(n+2))^n)
-
Similar expressions
- 1/((ln(n-2))^n)
Sum of series 1/((ln(n+2))^n)
The solution
You have entered
[src]
oo ____ \ ` \ 1 \ ----------- / n / log (n + 2) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\log{\left(n + 2 \right)}^{n}}$$
Sum(1/(log(n + 2)^n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\log{\left(n + 2 \right)}^{n}}$$
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
$$a_{n} = \log{\left(n + 2 \right)}^{- n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\log{\left(n + 2 \right)}^{- n} \log{\left(n + 3 \right)}^{n + 1}\right)$$
Let's take the limit
we find
$$\frac{1}{\log{\left(n + 2 \right)}^{n}}$$
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
$$a_{n} = \log{\left(n + 2 \right)}^{- n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\log{\left(n + 2 \right)}^{- n} \log{\left(n + 3 \right)}^{n + 1}\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ -n / log (2 + n) /__, n = 1
$$\sum_{n=1}^{\infty} \log{\left(n + 2 \right)}^{- n}$$
Sum(log(2 + 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