Sum of series exp(-n*b)

You have entered [src]

  oo       
 ___       
 \  `      
  \    -n*b
  /   e    
 /__,      
n = 1

\sum_{n=1}^{\infty} e^{b \left(- n\right)}

Sum(exp((-n)*b), (n, 1, oo))

The radius of convergence of the power series

Given number:

e^{b \left(- n\right)}

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 = - b

c = 0

then

R^{- b} = \tilde{\infty} \left(- e + \lim_{n \to \infty} 1\right)

False

R = \tilde{\infty}^{- \frac{1}{b}}

The answer [src]

  oo       
 ___       
 \  `      
  \    -b*n
  /   e    
 /__,      
n = 1

\sum_{n=1}^{\infty} e^{- b n}

Sum(exp(-b*n), (n, 1, oo))