proof of Abel’s lemma (by induction)
Proof. The proof is by induction. However, let us first recall that sum on the right side is a piece-wise defined function of the upper limit . In other words, if the upper limit is below the lower limit , the sum is identically set to zero. Otherwise, it is an ordinary sum. We therefore need to manually check the first two cases. For the trivial case , both sides equal to . Also, for (when the sum is a normal sum), it is easy to verify that both sides simplify to . Then, for the induction step, suppose that the claim holds for some . For , we then have
Since , the claim follows. .
|Title||proof of Abel’s lemma (by induction)|
|Date of creation||2013-03-22 13:38:04|
|Last modified on||2013-03-22 13:38:04|
|Last modified by||mathcam (2727)|