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Sum of series:
x^n/n
1/(4n^2-1)
(1000^n)/n!
2n
Identical expressions
(x+ three)^n/ two ^n
(x plus 3) to the power of n divide by 2 to the power of n
(x plus three) to the power of n divide by two to the power of n
(x+3)n/2n
x+3n/2n
x+3^n/2^n
(x+3)^n divide by 2^n
Similar expressions
(x-3)^n/2^n

Sum of series (x+3)^n/2^n

The solution

You have entered [src]

  oo          
____          
\   `         
 \           n
  \   (x + 3) 
   )  --------
  /       n   
 /       2    
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{\left(x + 3\right)^{n}}{2^{n}}$$

Sum((x + 3)^n/2^n, (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\left(x + 3\right)^{n}}{2^{n}}$$
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
$$a_{n} = 2^{- n}$$
and
$$x_{0} = -3$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = -3 + \lim_{n \to \infty}\left(2^{- n} 2^{n + 1}\right)$$
Let's take the limit
we find
$$R = -1$$

The answer [src]

/      3   x                        
|      - + -                        
|      2   2             |3   x|    
|     -------        for |- + -| < 1
|       1   x            |2   2|    
|     - - - -                       
|       2   2                       
<                                   
|  oo                               
| ___                               
| \  `                              
|  \    -n        n                 
|  /   2  *(3 + x)      otherwise   
| /__,                              
\n = 1

$$\begin{cases} \frac{\frac{x}{2} + \frac{3}{2}}{- \frac{x}{2} - \frac{1}{2}} & \text{for}\: \left|{\frac{x}{2} + \frac{3}{2}}\right| < 1 \\\sum_{n=1}^{\infty} 2^{- n} \left(x + 3\right)^{n} & \text{otherwise} \end{cases}$$

Piecewise(((3/2 + x/2)/(-1/2 - x/2), |3/2 + x/2| < 1), (Sum(2^(-n)*(3 + 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
```

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Identical expressions

Similar expressions

Sum of series (x+3)^n/2^n

The solution

Examples of finding the sum of a series