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)^n/3^n|
-
sqrt(n^2+3n)-sqrt(n^2-n)
-
sqrt(n)/(n+3)
-
factorial(2*n)/4^n
-
Identical expressions
- |(- one)^n/ three ^n|
- module of ( minus 1) to the power of n divide by 3 to the power of n|
- module of ( minus one) to the power of n divide by three to the power of n|
- |(-1)n/3n|
- |-1n/3n|
- |-1^n/3^n|
- |(-1)^n divide by 3^n|
-
Similar expressions
- |(1)^n/3^n|
Sum of series |(-1)^n/3^n|
The solution
You have entered
[src]
oo ____ \ ` \ | n| \ |(-1) | ) |-----| / | n | / | 3 | /___, n = 1
$$\sum_{n=1}^{\infty} \left|{\frac{\left(-1\right)^{n}}{3^{n}}}\right|$$
Sum(Abs((-1)^n/3^n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left|{\frac{\left(-1\right)^{n}}{3^{n}}}\right|$$
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
$$a_{n} = 3^{- \operatorname{re}{\left(n\right)}} e^{- \pi \operatorname{im}{\left(n\right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{- n} 3^{n + 1}\right)$$
Let's take the limit
we find
$$\left|{\frac{\left(-1\right)^{n}}{3^{n}}}\right|$$
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
$$a_{n} = 3^{- \operatorname{re}{\left(n\right)}} e^{- \pi \operatorname{im}{\left(n\right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{- n} 3^{n + 1}\right)$$
Let's take the limit
we find
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