Infinite Geometric Series
Understand convergence and find limiting sums for geometric series.
An infinite geometric series can have a finite sum only when the terms get smaller and smaller.
Convergence condition and sum\[|r|\lt 1\qquad\Rightarrow\qquad S_\infty=\frac{u_1}{1-r}\]
Worked example
Find \(3+1.5+0.75+\cdots\).
Here \(u_1=3\) and \(r=0.5\).
\[S_\infty=\frac{3}{1-0.5}=6\]Interpretation
The partial sums approach 6 but never need to have a final term. The value 6 is a limit.