proof of identity theorem of power series
We can prove the identity theorem for power series
using divided differences
. From amongst the points
at which the two series are equal, pick a sequence
which satisfies the following
three conditions:
-
1.
-
2.
if and only if .
-
3.
for all .
Let be the function determined by one power series
and let be the function determined by the other
power series:
Because formation of divided differences involves finite sums and dividing by differences of ’s (which all differ from zero by condition 2 above, so it is legitimate to divide by them), we may carry out the formation of finite diffferences on a term-by-term basis. Using the result about divided differences of powers, we have
where
Note that when , but . Since power series converge uniformly, we may intechange limit and summation to conclude
Since, by design, , we have
hence for all .
Title | proof of identity theorem of power series |
---|---|
Canonical name | ProofOfIdentityTheoremOfPowerSeries1 |
Date of creation | 2013-03-22 16:48:46 |
Last modified on | 2013-03-22 16:48:46 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 11 |
Author | rspuzio (6075) |
Entry type | Proof |
Classification | msc 40A30 |
Classification | msc 30B10 |