Introduction to Series: Geometric and Telescoping
Define an infinite series as a limit of partial sums. Determine convergence and find the sum of geometric series |r| < 1 and telescoping series. Apply to repeating decimals and physical problems (Zeno's paradox).
A lesson of AP Calculus BC, in Infinite Sequences and Series, about 10 minutes.
Introduction to Series
By the end of this lesson, you will be able to define an infinite series through partial sums, determine convergence of geometric and telescoping series.
Infinite Series
An infinite series is the sum of all terms of a sequence: $$\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$$
Partial sums: $S_N = \sum_{n=1}^N a_n$
The series converges to $S$ if $\lim_{N\to\infty} S_N = S$ (finite). Otherwise it diverges.
Geometric Series
$$\sum_{n=0}^\infty ar^n = a + ar + ar^2 + \cdots$$
Converges if $|r| < 1$: $S = \frac{a}{1-r}$
Diverges if $|r| \geq 1$
Example: $\sum_{n=0}^\infty 3\left(\frac{1}{2}\right)^n = \frac{3}{1-1/2} = 6$
Finding the sum from partial sums: $S_N = a\cdot\frac{1-r^N}{1-r}$. As $N\to\infty$ with $|r|<1$: $r^N\to 0$ → $S = \frac{a}{1-r}$.
Telescoping Series
Terms cancel pairwise. Recognize by partial fraction decomposition.
Example: $\sum_{n=1}^\infty \frac{1}{n(n+1)}$
$\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$
$S_N = \left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\cdots+\left(\frac{1}{N}-\frac{1}{N+1}\right) = 1 - \frac{1}{N+1} \to 1$
The series converges to 1.
Summary
Infinite series: defined by its partial sums; converges if the partial sums converge.
Geometric series: $\sum ar^n$ converges to $\frac{a}{1-r}$ iff $|r|<1$.
Telescoping: partial sums collapse; find $\lim S_N$ directly.