Mister Exam

Lang:

Other calculators

How to use it?
Sum of series:
(3^n-1)/n!
sin2n
(5^n-4^n)/6^n
sqrt(n^3+2)/(n^2+3)
Identical expressions
((nine x- one)^n)/((n+ one)*(9^n))
((9x minus 1) to the power of n) divide by ((n plus 1) multiply by (9 to the power of n))
((nine x minus one) to the power of n) divide by ((n plus one) multiply by (9 to the power of n))
((9x-1)n)/((n+1)*(9n))
9x-1n/n+1*9n
((9x-1)^n)/((n+1)(9^n))
((9x-1)n)/((n+1)(9n))
9x-1n/n+19n
9x-1^n/n+19^n
((9x-1)^n) divide by ((n+1)*(9^n))
Similar expressions
((9x-1)^n)/((n-1)*(9^n))
((9x+1)^n)/((n+1)*(9^n))

Sum of series ((9x-1)^n)/((n+1)*(9^n))

The solution

You have entered [src]

  oo            
____            
\   `           
 \             n
  \   (9*x - 1) 
   )  ----------
  /            n
 /    (n + 1)*9 
/___,           
n = 1

$$\sum_{n=1}^{\infty} \frac{\left(9 x - 1\right)^{n}}{9^{n} \left(n + 1\right)}$$

Sum((9*x - 1)^n/(((n + 1)*9^n)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\left(9 x - 1\right)^{n}}{9^{n} \left(n + 1\right)}$$
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
$$a_{n} = \frac{9^{- n}}{n + 1}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 9$$
then
$$R = \frac{1 + \lim_{n \to \infty}\left(\frac{9^{- n} 9^{n + 1} \left(n + 2\right)}{n + 1}\right)}{9}$$
Let's take the limit
we find
$$R = \frac{10}{9}$$

The answer [src]

//  1    x\ /     2       2*log(10/9 - x)\                              
||- -- + -|*|- -------- - ---------------|  for And(x >= -8/9, x < 10/9)
|\  18   2/ |  -1/9 + x               2  |                              
|           \               (-1/9 + x)   /                              
|                                                                       
|            oo                                                         
|          ____                                                         
<          \   `                                                        
|           \     -n           n                                        
|            \   9  *(-1 + 9*x)                                         
|            /   ---------------                     otherwise          
|           /         1 + n                                             
|          /___,                                                        
|          n = 1                                                        
\

$$\begin{cases} \left(\frac{x}{2} - \frac{1}{18}\right) \left(- \frac{2}{x - \frac{1}{9}} - \frac{2 \log{\left(\frac{10}{9} - x \right)}}{\left(x - \frac{1}{9}\right)^{2}}\right) & \text{for}\: x \geq - \frac{8}{9} \wedge x < \frac{10}{9} \\\sum_{n=1}^{\infty} \frac{9^{- n} \left(9 x - 1\right)^{n}}{n + 1} & \text{otherwise} \end{cases}$$

Piecewise(((-1/18 + x/2)*(-2/(-1/9 + x) - 2*log(10/9 - x)/(-1/9 + x)^2), (x >= -8/9)∧(x < 10/9)), (Sum(9^(-n)*(-1 + 9*x)^n/(1 + 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
```

Other calculators

How to use it?

Identical expressions

Similar expressions

Sum of series ((9x-1)^n)/((n+1)*(9^n))

The solution

Examples of finding the sum of a series