8.2. Infinite Series http://www.ck12.org
8.2 Infinite Series
Learning Objectives
- Demonstrate an understanding of series and the sequence of partial sums
- Recognize geometric series and determine when they converge or diverge
- Compute the sum of a convergent geometric series
- Determine convergence or divergence of series using the nth-Term Test
Infinite Series (series, sequence of partial sums, convergence, divergence)
Series
Another topic that involves an infinite number of terms is the topic ofinfinite series. We can represent certain
functions and numbers with an infinite series. For example, any real number that can be written as a non-terminating
decimal can be represented as an infinite series.
Example 1
The rational number^49 can be written as 0.44444.... We can expand the decimal notation as an infinite series:
4
9 =^0.^4 +^0.^04 +^0.^004 +^0.^0004 +...
= 104 + 1004 + 10004 + 10 ,^4000 +...
= 104 + 1042 + 1043 + 1044 +...
On the other hand, the number^14 can be written as 0.25. If we expand the decimal notation, we get a finite series:
1
4 =^0.^2 +^0.^05
= 102 + 1005
= 102 + 1052
Do you see the difference between an infinite series and a finite series? Let’s define what we mean by aninfinite
series.
Infinite Series
An infinite series is the sum of an infinite number of terms,u 1 ,u 2 ,u 3 ,u 4 ,...,usually written as.