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:
- ln^n(x)/n!
-
(3^n-1)/factorial(n)
-
ln(n/10)/(exp(n/10)-1)
-
lnx
- Integral of d{x}:
- lnx
- Inequation:
- lnx
- Graphing y =:
- lnx
-
Identical expressions
- lnx
-
Similar expressions
- ln(x)
- 1/(x*ln(x)^2)
- 1/(ln(x+2)*ln(x+2))
- ln(x)/x
Sum of series lnx
The solution
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oo __ \ ` ) log(x) /_, x = 1
$$\sum_{x=1}^{\infty} \log{\left(x \right)}$$
Sum(log(x), (x, 1, oo))
The radius of convergence of the power series
Given number:
$$\log{\left(x \right)}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \log{\left(x \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{\left|{\log{\left(x \right)}}\right|}{\log{\left(x + 1 \right)}}\right)$$
Let's take the limit
we find
$$\log{\left(x \right)}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \log{\left(x \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{\left|{\log{\left(x \right)}}\right|}{\log{\left(x + 1 \right)}}\right)$$
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