Mister Exam
Other calculators
1/n!
1/(2^n)
1/((4n-3)*(4n+1))
Sum of series x(1-x)^n
The solution
You have entered
[src]
oo ___ \ ` \ n / x*(1 - x) /__, n = 1
$$\sum_{n=1}^{\infty} x \left(1 - x\right)^{n}$$
Sum(x*(1 - x)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$x \left(1 - x\right)^{n}$$
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
$$a_{n} = x$$
and
$$x_{0} = -1$$
,
$$d = 1$$
,
$$c = -1$$
then
$$R = - (-1 + \lim_{n \to \infty} 1)$$
Let's take the limit
we find
$$R = 0$$
$$x \left(1 - x\right)^{n}$$
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
$$a_{n} = x$$
and
$$x_{0} = -1$$
,
$$d = 1$$
,
$$c = -1$$
then
$$R = - (-1 + \lim_{n \to \infty} 1)$$
Let's take the limit
we find
$$R = 0$$
The answer
[src]
// 1 - x \ || ----- for |-1 + x| < 1| || x | || | || oo | x*|< ___ | || \ ` | || \ n | || / (1 - x) otherwise | || /__, | \\n = 1 /
$$x \left(\begin{cases} \frac{1 - x}{x} & \text{for}\: \left|{x - 1}\right| < 1 \\\sum_{n=1}^{\infty} \left(1 - x\right)^{n} & \text{otherwise} \end{cases}\right)$$
x*Piecewise(((1 - x)/x, |-1 + x| < 1), (Sum((1 - x)^n, (n, 1, oo)), True))
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