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:
-
3n
- nx^n!
-
(3-k)/(k*(k+3)*(k+1))
-
sqrt(n)sin(1/(2n))^2
-
Identical expressions
- three (four / three)^(n- one)
- 3(4 divide by 3) to the power of (n minus 1)
- three (four divide by three) to the power of (n minus one)
- 3(4/3)(n-1)
- 34/3n-1
- 34/3^n-1
- 3(4 divide by 3)^(n-1)
-
Similar expressions
- 3(4/3)^(n+1)
Sum of series 3(4/3)^(n-1)
The solution
You have entered
[src]
oo ___ \ ` \ n - 1 / 3*4/3 /__, n = 0
$$\sum_{n=0}^{\infty} 3 \left(\frac{4}{3}\right)^{n - 1}$$
Sum(3*(4/3)^(n - 1), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$3 \left(\frac{4}{3}\right)^{n - 1}$$
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
$$a_{n} = 3 \left(\frac{4}{3}\right)^{n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{4}{3}\right)^{- n} \left(\frac{4}{3}\right)^{n - 1}\right)$$
Let's take the limit
we find
$$3 \left(\frac{4}{3}\right)^{n - 1}$$
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
$$a_{n} = 3 \left(\frac{4}{3}\right)^{n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{4}{3}\right)^{- n} \left(\frac{4}{3}\right)^{n - 1}\right)$$
Let's take the limit
we find
False
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