Which of the following?

Question

Which of the following is the sum of the series pi^n/e^(2n) from n=0 to infinity? 

 

a) e/(e-pi)

b) pi/(pi-e)

c) pi/(pi-e^2)

d) e^2/(e^2-pi)

e) The series diverges. 

 

Answer: D

 

Please show all your work. Thanks. 

Answer 1

S = ∑{n=0,∞}(pi^n/e^(2n))

 

pi^n/e^(2n) = (pi/e^2)^n

 

pi/e^2 ≈ 0.42516833158763 < 1

 

So pi^n/e^(2n) = (pi/e^2)^n → 0 as n → ∞

and S converges.

 

S = 1 + pi/e^2 + pi^2/e^4 + pi^3/e^6 + ...

 

S pi/e^2 = pi/e^2 + pi^2/e^4 + pi^3/e^6 + ...

 

S – S pi/e^2 = 1

 

S = 1/(1 – pi/e^2) = e^2/(e^2 – pi)

 

Answer is D.