Mister Exam

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Sum of series:
(-1)^(n+1)3^(2n)x^(2n)
cos((n*pi)/180)
24000
0.79*10^-6
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!)
Similar expressions
((-1)^(x(x+1)/2))/(x!)
((1)^(x(x-1)/2))/(x!)

Sum of series/ ((-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

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))

Numerical answer [src]

1.38177329067603622405343892907

1.38177329067603622405343892907

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
```

Other calculators

How to use it?

Identical expressions

Similar expressions

Sum of series ((-1)^(x(x-1)/2))/(x!)

The solution

Examples of finding the sum of a series