6.4 Introduction to Series
A series is "a sum of the terms of a sequence". Simply put, it's a sequence that is seperated by addition signs, not commas.
Example: 4+6+8+10+12 ...
- Series may be described by employing the letter Sigma derived from the Greek alphabet.
6.5 Type 1: Arithmetic Series
A series in which the difference between the consecutive terms if consistent.
Ex. 1+3+5+7+9...
*Sn is the sum of the terms in a series
Sn = n/2 (a + tn)
It's all based on Carl F. Gauss' theorem devised at the age of 8. Gauss was punished by his teacher and was instructed to add from 1 to 100. It took him only a moment, since he added the original series (1+2+...100) with its reverse (100+99+...1), multiplied it by the number of terms there are (ie 101) and divided by two.
Ex. "Find the sum of the first 50 terms for -10-12-14..."
Solution:
"S50 = 50/2 [(2(-10) + (50-1) (-2)]
S50= -2950"
Therefore, the sum of the first 50 terms is -2950.
6.6 Type 2: Geometric Series
A series in which the ratio between the consecutive terms is consistent.
Formula:
Sn = a(r^n -1) / r-1
Example: Find the sum of the first ten terms for the series 1+3+9+27...
S10 = 1(3^10-1) / 3-1
S10 = 59048/2
S10 = 29524
Therefore, the sum of the first 10 terms is 29524.