Mister Exam

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Sum of series/ exp(-0.01*n)

Sum of series exp(-0.01*n)

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  oo      
____      
\   `     
 \     -n 
  \    ---
  /    100
 /    e   
/___,     
n = 1

\sum_{n=1}^{\infty} e^{- \frac{n}{100}}

Sum(exp(-n/100), (n, 1, oo))

The radius of convergence of the power series

Given number:

e^{- \frac{n}{100}}

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} = - e

d = - \frac{1}{100}

c = 0

then

\frac{1}{\sqrt[100]{R}} = \tilde{\infty} \left(- e + \lim_{n \to \infty} 1\right)

False

R = 0

The rate of convergence of the power series

The answer [src]

   -1/100  
  e        
-----------
     -1/100
1 - e

\frac{1}{\left(1 - e^{- \frac{1}{100}}\right) e^{\frac{1}{100}}}

exp(-1/100)/(1 - exp(-1/100))

Numerical answer [src]

99.5008333319444477513144841479

99.5008333319444477513144841479

The graph