Series Definition of a Series Example Consider the sequence         an  =  n2           {1, 4, 9, 16, 25, ...}  We make the following definition.           S1  = a1  =  1         S2  =  a1 + a2  =  1 + 5  =  5         S3  =  a1 + a2 + a3  =  1 + 4 + 9  =  13   Exercise  Find S4  and S5    In general., for a sequence {an}  we define a new sequence called the sequence of partial sums by         Sn =  a1 + a2 + a3 + ... + an     ExerciseFind S5 for an  =  2n + 1  an  =  (-1)n   Sigma Notation Instead of using the " ... " notation we use the following notation: Example =  1/3 + 1/4 + 1/5 + 1/6 We read this as  "The sum from i equals 3 to 6 of 1 over i."   i is called the index of summation.  Think of sigma as a big plus sign.  The bottom number tells you where to start and the top number tells you where to end.   Example = (1 + 4) + (1 + 9) + (1 + 16) + ( 1 + 25) + (1 + 36)   Example Write          3 + 5 + 7 + 9 + ... + 23 in sigma notation   Solution Step 1  Identify an  (this goes to the right of the sigma sign) We see that          an  =  2n + 1 Step 2  Solve for n to find out what n the first term uses.  (this goes on the bottom of the sigma sign) Since          2n+ 1 = 3   has solution          n = 1 the first term uses n = 1 Step 3  Find out what n the last term uses (this goes on the top of the sigma sign) We solve:          2n + 1 = 23  has solution          n = 12 Step 4  Write             for the example,         Exercises Write the following in sigma notation  {3, 6, 9, 12, ... 120} {-1, 2, -4, ..., 128}   Back to the Intermediate Algebra (Math 154) Home Page