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:
-
(3/4)^n
-
1/((2n-1)*(2n+1))
-
1/(n*(n+3))
-
(5/6)^n
- Derivative of:
-
(x-1)²
- Integral of d{x}:
-
(x-1)²
-
Identical expressions
- (x- one)²
- (x minus 1)²
- (x minus one)²
- x-1²
-
Similar expressions
- (x+1)²
Sum of series (x-1)²
The solution
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oo ___ \ ` \ 2 / (x - 1) /__, x = 1
$$\sum_{x=1}^{\infty} \left(x - 1\right)^{2}$$
Sum((x - 1)^2, (x, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(x - 1\right)^{2}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \left(x - 1\right)^{2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{\left(x - 1\right)^{2}}{x^{2}}\right)$$
Let's take the limit
we find
$$\left(x - 1\right)^{2}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \left(x - 1\right)^{2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{\left(x - 1\right)^{2}}{x^{2}}\right)$$
Let's take the limit
we find
True
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