Three Sequences with Limit e

choose ϵ > 0

0.0565

find m

95

find k

5

zoom

0.599

This Demonstration illustrates three convergent sequences with limit

e

:

a

n

=

n

(1+1/n)

,

b

n

=

-n

(1-1/n)

, and

c

n

=1+1/1!+1/2!+..+1/n!

. The first two converge slowly and the last one converges quickly. Use the second slider to choose

m

so that if

n≥m

, then

a

n

and

b

n

are within

ϵ

of

e

. Use the third slider to choose

k

so that if

n≥k

, then

c

n

is within

ϵ

of

e

.