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+(-3)^n)/6^n
-
1/((2n-1)(2n+5))
- sin(pi*sqrt(n^2+k^2))
-
ln(n/10)/(exp(n/10)+1)
-
Identical expressions
- ln(n/ ten)/(exp(n/ ten)+ one)
- ln(n divide by 10) divide by ( exponent of (n divide by 10) plus 1)
- ln(n divide by ten) divide by ( exponent of (n divide by ten) plus one)
- lnn/10/expn/10+1
- ln(n divide by 10) divide by (exp(n divide by 10)+1)
-
Similar expressions
- ln(n/10)/(exp(n/10)-1)
Sum of series ln(n/10)/(exp(n/10)+1)
The solution
You have entered
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oo
______
\ `
\ /n \
\ log|--|
\ \10/
\ -------
/ n
/ --
/ 10
/ e + 1
/_____,
n = 1
$$\sum_{n=1}^{\infty} \frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} + 1}$$
Sum(log(n/10)/(exp(n/10) + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} + 1}$$
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
$$a_{n} = \frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(e^{\frac{n}{10} + \frac{1}{10}} + 1\right) \left|{\frac{\log{\left(\frac{n}{10} \right)}}{\log{\left(\frac{n}{10} + \frac{1}{10} \right)}}}\right|}{e^{\frac{n}{10}} + 1}\right)$$
Let's take the limit
we find
$$\frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} + 1}$$
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
$$a_{n} = \frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(e^{\frac{n}{10} + \frac{1}{10}} + 1\right) \left|{\frac{\log{\left(\frac{n}{10} \right)}}{\log{\left(\frac{n}{10} + \frac{1}{10} \right)}}}\right|}{e^{\frac{n}{10}} + 1}\right)$$
Let's take the limit
we find
False
False
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