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(1+0.9)^(1/n)
Sum of series/ (1+0.9)^(1/n)

Sum of series (1+0.9)^(1/n)


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The solution

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  oo              
 ___              
 \  `             
  \   n __________
  /   \/ 9/10 + 1 
 /__,             
n = 1
$$\sum_{n=1}^{\infty} \left(\frac{9}{10} + 1\right)^{\frac{1}{n}}$$ 
Sum((9/10 + 1)^(1/n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{9}{10} + 1\right)^{\frac{1}{n}}$$
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
$$a_{n} = \left(\frac{19}{10}\right)^{\frac{1}{n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{19}{10}\right)^{\frac{1}{n}} \left(\frac{19}{10}\right)^{- \frac{1}{n + 1}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo          
____          
\   `         
 \        ____
  \      / 19 
  /   n /  -- 
 /    \/   10 
/___,         
n = 1
$$\sum_{n=1}^{\infty} \left(\frac{19}{10}\right)^{\frac{1}{n}}$$ 
Sum((19/10)^(1/n), (n, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series (1+0.9)^(1/n)

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