We have (1).
Multiplying (1) with we have
Substracting (2) from (1) yields
Since you answered so quickly, here is a second challenge. I am not sure if is a fair question to ask but I post it anyway.
Having shown that
Can you prove the following?
Let with radius of convergence
Then, is differenciable on and furthermore .
Note: This works even if is a complex number, but the proofs are completely anagolous.
By my Complex Analysis book had a really nice approach to this proof. Without using more advandeced techiqnues. It was able to prove term-by-term differenciation by using the series posted above. It was a long but straightforward derivation. That is what I am asking to show.