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:
if and only if .
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
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|
|Date of creation||2013-03-22 16:48:46|
|Last modified on||2013-03-22 16:48:46|
|Last modified by||rspuzio (6075)|