Mister Exam

### Other calculators

• #### How to use it?

• Sum of series:
• x^(n)
• 1\3^n
• x^(4n)/n!
• 1/n*(n+1)(n+2)
• Integral of d{x}:
• x^(n)
• #### Identical expressions

• x^(n)
• x to the power of (n)
• x(n)
• xn
• x^n

# Sum of series x^(n)

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### The solution

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  oo
___
\
\    n
/   x
/__,
n = 1   
$$\sum_{n=1}^{\infty} x^{n}$$
Sum(x^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$x^{n}$$
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
$$a_{n} = 1$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty} 1$$
Let's take the limit
we find
$$R = 1$$
/   x
| -----    for |x| < 1
| 1 - x
|
|  oo
< ___
| \  
|  \    n
|  /   x    otherwise
| /__,
\n = 1                
$$\begin{cases} \frac{x}{1 - x} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=1}^{\infty} x^{n} & \text{otherwise} \end{cases}$$
Piecewise((x/(1 - x), |x| < 1), (Sum(x^n, (n, 1, oo)), True))
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