Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
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How to use it?
- Sum of series:
-
6/((3n-2)(3n+4))
-
(n-1)/2^(n+1)
-
1/((4n-3)(4n+1))
- sen(ix)
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Identical expressions
- ((- one)^(x(x- one)/ two))/(x!)
- (( minus 1) to the power of (x(x minus 1) divide by 2)) divide by (x!)
- (( minus one) to the power of (x(x minus one) divide by two)) divide by (x!)
- ((-1)(x(x-1)/2))/(x!)
- -1xx-1/2/x!
- -1^xx-1/2/x!
- ((-1)^(x(x-1) divide by 2)) divide by (x!)
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Similar expressions
- ((-1)^(x(x+1)/2))/(x!)
- ((1)^(x(x-1)/2))/(x!)
Sum of series ((-1)^(x(x-1)/2))/(x!)
The solution
You have entered
[src]
oo _____ \ ` \ x*(x - 1) \ --------- \ 2 / (-1) / ------------- / x! /____, x = 0
$$\sum_{x=0}^{\infty} \frac{\left(-1\right)^{\frac{x \left(x - 1\right)}{2}}}{x!}$$
Sum((-1)^((x*(x - 1))/2)/factorial(x), (x, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{\frac{x \left(x - 1\right)}{2}}}{x!}$$
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
$$a_{x} = \frac{\left(-1\right)^{\frac{x \left(x - 1\right)}{2}}}{x!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty} \left|{\frac{\left(x + 1\right)!}{x!}}\right|$$
Let's take the limit
we find
$$\frac{\left(-1\right)^{\frac{x \left(x - 1\right)}{2}}}{x!}$$
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
$$a_{x} = \frac{\left(-1\right)^{\frac{x \left(x - 1\right)}{2}}}{x!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty} \left|{\frac{\left(x + 1\right)!}{x!}}\right|$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
oo _____ \ ` \ x*(-1 + x) \ ---------- \ 2 / (-1) / -------------- / x! /____, x = 0
$$\sum_{x=0}^{\infty} \frac{\left(-1\right)^{\frac{x \left(x - 1\right)}{2}}}{x!}$$
Sum((-1)^(x*(-1 + x)/2)/factorial(x), (x, 0, oo))
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