Calculus II · Unit 4A · lesson
Geometric Series
Derive and use finite and infinite geometric-sum formulas, handle shifted indices, repeating decimals, and convergence conditions.
Section overview
Infinite series and foundational examplesWhat this section is building
Derive and use finite and infinite geometric-sum formulas, handle shifted indices, repeating decimals, and convergence conditions.
A series converges exactly when its sequence of partial sums approaches a finite value.
Check the term limit first, then look for an exact partial-sum pattern before selecting a comparison test.
Concluding convergence from terms approaching zero or canceling inside an unwritten infinite expression.
Learning objectives
recognize, test, and sum geometric series and interpret the common ratio.
Geometric Series
Repeated proportional change creates the simplest infinite series
A geometric series multiplies each term by the same ratio. That repeated scaling makes the partial sums algebraically manageable: multiplying the finite sum by the ratio lines up almost every term for cancellation. The resulting formula reveals both convergence and the exact sum.
The ratio controls everything. When , powers of shrink to zero and the partial sums settle. When , the terms do not approach zero, so the series cannot converge. In applications, the ratio may represent retained energy after each bounce, a repeated discount, a reflection coefficient, or the fraction of material remaining after each stage.
Repeated scaling creates an exact formula
A geometric series multiplies each term by a constant ratio . Multiplying a finite partial sum by shifts its terms, so subtraction leaves only the first term and a final remainder.
The condition means that remainder vanishes. When , the terms do not shrink to zero, so convergence is impossible.
Read this graph as text
The remaining tail is a scaled copy. Lengths one-half, one-quarter, one-eighth, and so on fill a unit interval. Self-similar interval filling and a shrinking tail. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in the remaining tail is a scaled copy; color is never the only cue.
Why it matters: Self-similar interval filling and a shrinking tail.
Lengths one-half, one-quarter, one-eighth, and so on fill a unit interval.
The remaining tail is a scaled copy. Self-similar interval filling and a shrinking tail.
Finite identity first, limit second
For , subtraction gives
Thus . Only after this finite formula is established do we let .
Read this graph as text
A geometric series repeatedly takes the same fraction. Successive pieces occupy one half of what remains, producing areas 1/2,1/4,1/8, whose total fills one whole square. Self-similar interval filling and a shrinking tail. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in a geometric series repeatedly takes the same fraction; color is never the only cue.
Why it matters: Self-similar interval filling and a shrinking tail.
Successive pieces occupy one half of what remains, producing areas whose total fills one whole square.
A geometric series repeatedly takes the same fraction. Self-similar interval filling and a shrinking tail.
The uncovered strip after each step is again half as wide as before. The self-similarity is the geometric ratio in visual form.
Geometric-series formula
For ,
If , the series diverges.
Sum a shifted geometric series
Consider
The first included term is , and the ratio is . Therefore
Using the actual first term avoids an indexing error.
A repeating decimal is a convergent series
The decimal equals
This is geometric with first term and ratio , so
Use the first included term
For a series starting at , the first term is obtained by substituting . A shifted index changes the first term even when the ratio is unchanged.
u4a-geometric_series-01Evaluate .
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Determine whether converges.
Write as a geometric series and fraction.
A ball rebounds to 70 percent of its previous height. Model total vertical distance after a drop from 10 meters.
Explain why the first term depends on the starting index.
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