Mister Exam

### Other calculators

• #### How to use it?

• Sum of series:
• (2^n+5^n)/10^n
• (x-1)^n/3^n
• a^n/factorial(n)
• (2^(n+2))/(7^n*9^(n-1))
• #### Identical expressions

• (x- one)^n/ three ^n
• (x minus 1) to the power of n divide by 3 to the power of n
• (x minus 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

• (x+1)^n/3^n

# Sum of series (x-1)^n/3^n

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### 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
$$R = 4$$
/        1   x
|      - - + -
|        3   3            |  1   x|
|      -------        for |- - + -| < 1
|       4   x             |  3   3|
|       - - -
|       3   3
<
|  oo
| ___
| \  
|  \    -n         n
|  /   3  *(-1 + x)       otherwise
| /__,
\n = 1                                 
$$\begin{cases} \frac{\frac{x}{3} - \frac{1}{3}}{\frac{4}{3} - \frac{x}{3}} & \text{for}\: \left|{\frac{x}{3} - \frac{1}{3}}\right| < 1 \\\sum_{n=1}^{\infty} 3^{- n} \left(x - 1\right)^{n} & \text{otherwise} \end{cases}$$
Piecewise(((-1/3 + x/3)/(4/3 - x/3), |-1/3 + x/3| < 1), (Sum(3^(-n)*(-1 + x)^n, (n, 1, oo)), True))