Mister Exam
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- 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/ln(5^n)
-
ln(5)^n/n!
-
(-1)^(n+1)/ln(n+2)
-
((n^2)^1/3)*atan(1/n^2)
-
Identical expressions
- ln(five)^n/n!
- ln(5) to the power of n divide by n!
- ln(five) to the power of n divide by n!
- ln(5)n/n!
- ln5n/n!
- ln5^n/n!
- ln(5)^n divide by n!
Sum of series ln(5)^n/n!
The solution
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oo ____ \ ` \ n \ log (5) / ------- / n! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\log{\left(5 \right)}^{n}}{n!}$$
Sum(log(5)^n/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\log{\left(5 \right)}^{n}}{n!}$$
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
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = - \log{\left(5 \right)}$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- \log{\left(5 \right)} + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
$$R = \infty$$
$$\frac{\log{\left(5 \right)}^{n}}{n!}$$
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
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = - \log{\left(5 \right)}$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- \log{\left(5 \right)} + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
$$R = \infty$$
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