CK-12-Calculus

8.1. Sequences http://www.ck12.org

Thus, limn→+∞ 0 ≤limn→+∞^8 nn!≤limn→+∞(^8 n)

( 87

7!

)

. By using the Rules Theorem, we have limn→+∞ 0 =0 and

limn→+∞(^8 n)

( 87

7!

)

=

( 87

7!

)

limn→+∞^8 n=

( 87

7!

)

× 0 =0. Thus, 0≤limn→+∞^8 nn!≤0. By the Sandwich/Squeeze
Theorem, limn→+∞^8 nn!=0.

Picard’s Method

The following method appeared in 1891 by Emile Picard, a famous French mathematician. It is a method for solving
initial value problems in differential equations that produces a sequence of functions which converge to the solution.
Start with the initial value problem:
y′=f(x,y)withy(x 0 ) =y 0
Iff(x,y)andfx(x,y)are both continuous then a unique solution to the initial value problem exists by Picard’s theory.
Now ify(x)is the solution to the given problem, then a reformulation of the differential equation is possible:

∫x

x 0 y

′(t)dt=∫x

x 0 f(t,y(t))dt

Now the Fundamental Theorem of Calculus is utilized to integrate the left hand side of the problem and upon
isolating, the following result is obtained:

y(x) =y 0 +

∫x

x 0 f(t,y(t))dt

The equation above is the starting point for the Picard iteration because it will be used to build the sequence of
functions which will describe the actual solution to the initial value problem. The Picard sequence of functions is
calculated as follows:
Step 1 - DefineY 0 (x) =y 0
Step 2 - SubstituteY 0 (t) =y 0 fory(t)inf(t,y(t)):

Y 1 (x) =y 0 +

∫x

x 0 f(t,Y^0 (t))dt

Y 1 (x) =y 0 +

∫x

x 0 f(t,Y^0 )dt

Step 3 - Repeat step 2 withY 1 (t)fory(t):

Y 2 (x) =y 0 +

∫x

x 0 f(t,Y^1 (t))dt

The substitution process is repeatedntimes and generates a sequence of functions{Yn(x)}which converges to the
initial value problem. To summarize this procedure mathematically,
Picard’s Method
Let{Yn(x)}be sequence defined successively by,