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Sum of series/ (1+5/n)^n

Sum of series (1+5/n)^n

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  oo          
____          
\   `         
 \           n
  \   /    5\ 
  /   |1 + -| 
 /    \    n/ 
/___,         
n = 1

\sum_{n=1}^{\infty} \left(1 + \frac{5}{n}\right)^{n}

Sum((1 + 5/n)^n, (n, 1, oo))

The radius of convergence of the power series

Given number:

\left(1 + \frac{5}{n}\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} = \left(1 + \frac{5}{n}\right)^{n}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\left(1 + \frac{5}{n}\right)^{n} \left(1 + \frac{5}{n + 1}\right)^{- n - 1}\right)

True

False

The rate of convergence of the power series

The graph