Suppose I have an equation of summations in the form:
$$\sum_{k=0}^{\infty}a_kz^k=\sum_{k=0}^{\infty}b_kz^k$$
Under what conditions and how can I deduce that: $$a_k=b_k\ \forall k$$
Thank you in advance for your answer
Asked
Active
Viewed 50 times
0
-
Always, as long as the equality holds for every $z$ in an interval $]-R,R[$, with $R>0$. – TheSilverDoe Jun 22 '23 at 18:04
-
It's called the identity theorem for complex power series. There is a maximal radius $r$ such that for all $|z|<r$ the series converges absolutely. Inside the common circle of convergency sums can be combined termwise and $0(z) = \sum (a_k-b_k)z^k $ is the zero function with $r=\infty$. – Roland F Jun 22 '23 at 18:08
-
The corresp. coeffs are equal for all $k \in \Bbb{N}$ if you deal in formal power series. – Daniel Donnelly Jun 22 '23 at 18:16
-
the set ${z^k}_k$ forms a basis within the function space - in fact this is the idea behind the Taylor series of a function – Dr. Richard Klitzing Jun 22 '23 at 19:04