data structures - Hashing analysis in hashtable -

the search time hash value o(1+alpha) ,

 alpha = number of elements/size of table 

i don't understand why 1 added?

the expected number elements examined

(1/n  summation of i=1 n (1+(i-1/m))) 

i don't understand too.how derived?

(i know how solve above expression , want understand how has been lead expression..)

edit : n number of elements present , m number of slots or size of table

i don't understand why 1 added?

the o(1) there tell if there no element in bucket or hash table @ all, you'll have compute key hash value , won't instantaneous.

your second part needs precisions. see comments.

edit: second portion there "amortized analysis", idea consider each insertion in fact in set of n insertions in empty hash table, each lookup take o(1) hashing plus o(i-1/m) searching bucket content considering each bucket evenly filled respect previous elements. resolution of sum gives o(1+alpha) amortized time.