#
# Written by:
# -- 
# John L. Weatherwax                2009-04-21
# 
# email: wax@alum.mit.edu
# 
# Please send comments and especially bug reports to the
# above email address.
#
#-----

kth_derivative_of_f = function(x,k){
  stopifnot( k>=0 )
  if( k==0 ){
    return( x^(1/3) )
  }else if ( k==1 ){
    return( x^(-2/3)/3 )
  }else {
    numer = 1
    term = 2
    while( term <= 3*k-4 ){
      numer = numer * term
      term = term + 3
    }
    denom = 3^k
    return( ( (-1)^(k-1) * numer / denom ) * x^( -(3*k-1)/3 ) )
  }
}

# Print the Taylor series terms: 
#
taylor_terms = c()
for( k in 0:5 ){
     tt = (-1)^k * kth_derivative_of_f( 8, k )/factorial(k) # note that (x-x_0)^k = (7-8)^k = (-1)^k
     taylor_terms = c( taylor_terms, tt )
     print( sprintf( "k= %5d, derivative= %10.6f", k, tt ) )
}
print( "taylor_terms= " ) 
print( taylor_terms ) 
print( "cumsum(taylor_terms)= " ) 
print( cumsum(taylor_terms) )