Mister Exam
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How to use it?
- Sum of series:
- x^n/n
-
1/n^6
-
1/n^n
-
(n-1)/n!
-
Identical expressions
- ln(one +(one /n))- one /(n+ one)
- ln(1 plus (1 divide by n)) minus 1 divide by (n plus 1)
- ln(one plus (one divide by n)) minus one divide by (n plus one)
- ln1+1/n-1/n+1
- ln(1+(1 divide by n))-1 divide by (n+1)
-
Similar expressions
- ln(1-(1/n))-1/(n+1)
- ln(1+(1/n))+1/(n+1)
- ln(1+(1/n))-1/(n-1)
Sum of series ln(1+(1/n))-1/(n+1)
The solution
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oo ___ \ ` \ / / 1\ 1 \ ) |log|1 + -| - -----| / \ \ n/ n + 1/ /__, n = 1
$$\sum_{n=1}^{\infty} \left(\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n + 1}\right)$$
Sum(log(1 + 1/n) - 1/(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n + 1}$$
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
$$a_{n} = \log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n + 1}}{\log{\left(1 + \frac{1}{n + 1} \right)} - \frac{1}{n + 2}}}\right|$$
Let's take the limit
we find
$$\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n + 1}$$
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
$$a_{n} = \log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n + 1}}{\log{\left(1 + \frac{1}{n + 1} \right)} - \frac{1}{n + 2}}}\right|$$
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