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Sum of series:
1
sin2n
2(x-6)^(4n)/(n(16)^n)
nx^n/(8n-9)*3^n
Identical expressions
(x+ one)^n/ three ^n
(x plus 1) to the power of n divide by 3 to the power of n
(x plus one) to the power of n divide by three to the power of n
(x+1)n/3n
x+1n/3n
x+1^n/3^n
(x+1)^n divide by 3^n
Similar expressions
(x-1)^n/3^n

Sum of series (x+1)^n/3^n

The solution

You have entered [src]

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

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

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

The radius of convergence of the power series

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

The answer [src]

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

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

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

The solution

Examples of finding the sum of a series