MY MATEX NOTES

Harmonic numbers Harmonic numbers are defined as and We can derive a recurrence relation to , namely Multiply both sides by Sum both sides from to infinity since we get (1) Sums involving Harmonic Numbers Now nefine (1) can be represented by (2) Note that if we differentiate w.r. to we get Multiplying both sides by On the other hand Therefore we get the recurrence relation (3) Lets now try to compute which corresponds to in the notation introduced above. From (3) we can work recursively to get (4) And from (2) we get that (5) Let start by computing the inner integral in (5), i.e. Doing a partial fraction we get The first integral we immediatly recognize it as . For the second one, lets compute the indefinite version first and then plug the limits to the result. First, lets make the change of variable . We get Performing a second change of variable, and therefore Plugging the limits In conclusion we get that (6) and fr