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Euler's Method for the Exponential Function

​
x
1.
approximation step
3
For each value of
x
, the sequence
e
n
(x)=
n
1+
x
n
converges to
x
e
. Also, if
E
n
(t)
is the Euler method approximation to the solution of the differential equation
y'=y
,
y(0)=1
on the interval
[0,x]
with
h=x/n
, then
E
n
(t)
(the red curve) converges to
t
e
for each
t
in the interval. The points generated by Euler's method are marked in green. Observe that for any
n
, the error between the approximation and the real value increases with the value of
x
.
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