Series
- Page ID
- 221346
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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
Exercise Find S5 for
Sigma Notation Instead of using the " ... " notation we use the following notation: Example \( \displaystyle\sum_{i=3}^{6} \frac{1}{i} \) = 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 \( \displaystyle\sum_{i=2}^{6} 1+i^2 \) = (1 + 4) + (1 + 9) + (1 + 16) + ( 1 + 25) + (1 + 36)
Example Write 3 + 5 + 7 + 9 + ... + 23 in sigma notation
Solution
Exercises Write the following in sigma notation
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