We give a approximate Sequence {Un(t

t0)} for U(t

t0)-operator

When n→∞

lt approach to Dyson's Series. We proof the following Lemma and theorems. Lemma. Suppose is a

set of operator functtions U(t) in Hilbert Space If (Ⅰ). When U (t) ∈ There is a Constant M

Such that‖U(t) ||1

t2

]. Then

there is a operator sequence U1~((1))(t)

U2~((2))(t)

……in Which is uniform We(?)k ConuergenCe on [t

t] With respects t0 t. Theorem I. If the norm ‖H(t)‖ of H(t) in eguation (2.1) is a Lebesgue integrablc function With respects to

t

Then the Dyson's series Converge Weakly

i.e. for any state Vector |φ>

|Ψ> Sequence'1|Ψ>

2Ψ>

......……

n|Ψ>

…… uniform Conuorgence With respects to t.Theorem 2. If in the finite time inte(?)val

The no(?)m ‖H(t)|| of H(t) i(?) a L(?)besgue integrable functien

Then equ(?)tion (2.1) has unique solution. The application to qu(?)tum mech(?)ics is discussed primarily.