chap analysis techniques from the formula for nt( )nt(ncn - (kk= ( ) ( ( ) (kk= subtracting nt(nfrom both sides yields( ) ( nt(nc( ) cn ( ( ) ( nt(nc( ( ( ) ( ( ( ) (nc( (nt( + + at this pointwe have eliminated the summation and can now use our normal methods for solving recurrences to get closed-form solution note that ( + + cso we can simplify the result expanding the recurrencewe get + (nn+ + + ( +  + + ( + - + ( ( ) + + + + + + **** + + +  ( ** + ( (hn+ ( < for hn+ the harmonic series from equation hn+ th(log )so the final solution is th( log