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