There are two questions in this
1) Remembering the reactivity series:
The word "reactivity series" is used in different ways in different classes. In class X it is used as the reactivity of the metal to displace the another metal in aqueous solution. As the reaction is in aqueous solution metals like Na, K are not considered in this series and only the series is Al>Zn>Fe>Cu etc.
In class XI it is used as the ability of the atom to donate or accept the electrons. So the alkali metals in the bottom of the series is more reactive and non metals in the top of the series are more reactive.
In class XII it is used as the electrochemical series, a series of electrode potentials of different elements and ions. There is no need to remember the entire series. It is not possible.
2) Remembering the complete periodic table:
This is also not required. When you are studying a certain part of periodic table the elements in that part will be remembered by you. For example when you are studying the properties of Nitrogen family you will remember the elements in that family (atleast first two and last one). In this way the number of elements remembered by you will grow and it is sufficient. There is no need to remember the entire periodic table.