The geometric series \(\sum \limits _{k= k_0}^\infty a \cdot r^k\)
- converges to \(\frac {a \, r^{k_0}}{1-r}\) when \(|r| < 1\).
- diverges if \(|r| \geq 1\).
In other words,
\(\blacktriangleright \) \(\sum \limits _{k= k_0}^\infty a \cdot r^k\) represents a number when \(|r| < 1\).
\(\blacktriangleright \) Otherwise, it does not represent a number.
That sounds like a function.
\[ \sum \limits _{k= k_0}^\infty a \cdot r^k \text { and } \frac {a \, r^{k_0}}{1-r} \text { represent the same function when } |r| < 1. \]
Learning Outcomes
In this this section, students will
- treat the geometric series as a function.
- obtain closed forms for geometric functions.
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more examples can be found by following this link
More Examples of Geometric Series
test applet...
2025-05-17 23:25:57