Mister Exam

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Sum of series:
(2/3)^n
3^n/n!
(3/7)^n
n/3^n
Limit of the function:
(2/3)^n
Identical expressions
(two / three)^n
(2 divide by 3) to the power of n
(two divide by three) to the power of n
(2/3)n
2/3n
2/3^n
(2 divide by 3)^n

Sum of series/ (2/3)^n

Sum of series (2/3)^n

The solution

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  oo      
 ___      
 \  `     
  \      n
  /   2/3 
 /__,     
n = 1

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

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

The radius of convergence of the power series

Given number:
$$\left(\frac{2}{3}\right)^{n}$$
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
$$x_{0} = - \frac{2}{3}$$
,
$$d = 1$$
,
$$c = 0$$
then