I'm studying this series: 

[![enter image description here][1]][1]


I know that there is this relation:
[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/V6yi0.jpg
  [2]: https://i.sstatic.net/l8TUU.jpg

(l is the common limit of the two series)

Do you know a mathematica function that could proove this relation? 

Thanks for your help!!