We have noted that decimal approximations or expansions can be viewed as a sequence of numbers, where a new digit is added for each term.

• \(4.1\)
• \(4.12\)
• \(4.123\)
• \(4.1231\)
• \(4.12310\)
• \(4.123105\)
• \(4.1231056\)
• \(4.12310562\)
• \(4.123105626\)

For decimal expansions, this pattern continues with no end.

We can view this decimal expansion in a different way: as a sum.

We would like to connect these two views.

One way is to make a sequence from the sum by adding on the next place value digit for each term:

Each term of the sequence is a part of the sum:

When summing up the terms in a sequence we obtain a series, a new way to represent numbers.

\(\blacktriangleright \) \(\sqrt {17} = 0.123105626\cdots \)

\(\blacktriangleright \) \(\sqrt {17} = \lim \limits _{\infty } \, \{ \frac {4}{10^0}, \frac {41}{10^1}, \frac {412}{10^2}, \frac {4123}{10^3}, \frac {41231}{10^4}, \frac {412310}{10^5}, \frac {4123105}{10^6}, \frac {41231056}{10^7}, \frac {412310562}{10^8}, \frac {4123105626}{10^9}, \cdots \}\)

\(\blacktriangleright \) \(\sqrt {17} = \lim \limits _{\infty } \, 4 + \frac {1}{10^1} + \frac {2}{10^2} + \frac {3}{10^3} + \frac {1}{10^4} + \frac {0}{10^5} + \frac {5}{10^6} + \frac {6}{10^7} + \frac {2}{10^8} + \frac {6}{10^9} + \cdots \)

Learning Outcomes

In this section, students will explore

• defining a series.
• recognizing a geometric series.
• recognizing a telescoping series.
• computing the sum of a geometric series.
• computing the sum of a telescoping series.