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:
-
(2/9)^n
-
sin(n*pi/2)/n
-
sin(n)^2/n^(3/2)
- e^ipi/n/n
-
Identical expressions
- (two / nine)^n
- (2 divide by 9) to the power of n
- (two divide by nine) to the power of n
- (2/9)n
- 2/9n
- 2/9^n
- (2 divide by 9)^n
Sum of series (2/9)^n
The solution
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oo ___ \ ` \ n / 2/9 /__, n = 1
$$\sum_{n=1}^{\infty} \left(\frac{2}{9}\right)^{n}$$
Sum((2/9)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{2}{9}\right)^{n}$$
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
$$a_{n} = 1$$
and
$$x_{0} = - \frac{2}{9}$$
,
$$d = 1$$
,
$$c = 0$$
then
Let's take the limit
we find
$$\left(\frac{2}{9}\right)^{n}$$
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
$$a_{n} = 1$$
and
$$x_{0} = - \frac{2}{9}$$
,
$$d = 1$$
,
$$c = 0$$
then
False
Let's take the limit
we find
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