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Sum of series:
((-1)^(n+1))/n
2^n/n^2
sin(n)
3^(n-1)/8^(n-1)
Identical expressions
exp(-n/ two)
exponent of ( minus n divide by 2)
exponent of ( minus n divide by two)
exp-n/2
exp(-n divide by 2)
Similar expressions
exp(n/2)

Sum of series/ exp(-n/2)

Sum of series exp(-n/2)

The solution

You have entered [src]

  oo      
____      
\   `     
 \     -n 
  \    ---
  /     2 
 /    e   
/___,     
n = 0

$$\sum_{n=0}^{\infty} e^{\frac{\left(-1\right) n}{2}}$$

Sum(exp((-n)/2), (n, 0, oo))

The radius of convergence of the power series

Given number:
$$e^{\frac{\left(-1\right) n}{2}}$$
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
$$x_{0} = - e$$
,
$$d = - \frac{1}{2}$$
,
$$c = 0$$
then
$$\frac{1}{\sqrt{R}} = \tilde{\infty} \left(- e + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find

False

$$R = 0$$

The rate of convergence of the power series

The answer [src]

    1    
---------
     -1/2
1 - e

$$\frac{1}{1 - e^{- \frac{1}{2}}}$$

1/(1 - exp(-1/2))

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
```

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Identical expressions

Similar expressions

Sum of series exp(-n/2)

The solution

Examples of finding the sum of a series