#lim_(n->oo)sum_(k=1)^n e^(-k)#?
1 Answer
Jun 4, 2017
#lim_(n->oo) sum_(k=1)^(n) e^(-k) = 1/(e-1)#
We can check our work here.
If we recognize that this can be rewritten as
#lim_(n->oo) sum_(k=1)^(n) (1/e)^k# ,
then this is related to the geometric series of the form
#sum_(k=0)^(oo) x^k = 1 + x + x^2 + . . . = 1 + sum_(k=1)^(oo) x^k = 1/(1-x)#
Therefore, let
#color(blue)(lim_(n->oo) sum_(k=1)^(n) (1/e)^k)#
#= 1/(1 - 1/e) - 1#
#= 1/((e - 1)/e) - 1#
#= e/(e-1) - 1#
#= (e - 1)/(e - 1) + 1/(e - 1) - 1#
#= color(blue)(1/(e-1))#
