Working with Geometric Series

3. Visual Connection

5. Concept Development

BC-only topic. A geometric series is one of the few infinite series in AP Calculus BC whose exact sum can often be found directly. Its special feature is a constant multiplier between consecutive nonzero terms.

Recognize geometric structure

A geometric series can be written as

where \(a\) is the first term and \(r\) is the common ratio. For explicit terms \(u_n\), verify

where the quotient is defined. A constant difference indicates an arithmetic sequence, not a geometric series.

Derive the finite partial-sum formula

For \(r\ne1\), let

Multiplying by \(r\) and subtracting cancels the interior terms:

This is a finite identity and is valid for any real \(r\ne1\). If \(r=1\), then \(S_N=Na\).

Pass from finite to infinite

The infinite series converges precisely when the partial sums approach a finite limit. Since \(r^N\to0\) exactly when \(|r|<1\),

The condition is part of the conclusion, not an optional check.

What happens when \(|r|\ge1\)?

• If \(r=1\) and \(a\ne0\), partial sums grow linearly.
• If \(r=-1\) and \(a\ne0\), partial sums oscillate.
• If \(|r|>1\), the terms do not approach zero and their magnitudes grow.

Except for the trivial case \(a=0\), the series diverges whenever \(|r|\ge1\).

Starting index determines the first term

In a sigma expression, substitute the lower index before using \(a/(1-r)\):

The coefficient \(c\) is not necessarily the first term. The first term is \(cr^m\).

Negative ratios

When \(-1<r<0\), terms alternate signs while shrinking in magnitude. Partial sums usually approach the sum from alternating sides. Keep the signed ratio in the denominator:

Tails and exact remainders

For \(\sum_{n=0}^{\infty}ar^n\), the tail beginning with exponent \(m\) is

If \(S_N=\sum_{n=0}^{N}ar^n\), the exact remainder after \(S_N\) is

This formula preserves the sign of an alternating remainder. Its absolute value gives the error magnitude.

Reindex without changing terms

The same series may appear as

When reindexing, adjust both the exponent and limits so the ordered list of terms remains unchanged.

Repeating decimals

A repeating block produces place values in a geometric pattern. For example, a two-digit repeating block has ratio \(10^{-2}\). Express the decimal as a series and sum it exactly rather than relying on rounded calculator output.

Repeated-process models

Rebounds, repeated reflections, retained medication, and recurring deposits can create geometric contributions. Define what one term measures, identify whether an initial quantity is counted once or twice, and check that a constant ratio is reasonable before summing.

Bridge to power series

The identity

will later generate many power-series representations through substitution, multiplication, differentiation, and integration.

Geometric-series checklist

1. Write several terms and verify a constant ratio.
2. Identify the actual first term from the lower index.
3. Distinguish a finite sum from an infinite sum.
4. For an infinite series, state and verify \(|r|<1\).
5. Use the signed ratio in \(a/(1-r)\).
6. Check the answer against the signs and approximate size of the first few partial sums.

6. Detailed Worked Example and Error Check

Example 1: An alternating geometric series

The series

has first term \(a=5\) and ratio \(r=-1/3\). Since \(|r|<1\),

Example 2: Starting index is not zero

For

the first term is \(3(1/2)^2=3/4\), not \(3\). Therefore

Example 3: Ratio \(-1\)

The series \(4-4+4-4+\cdots\) has \(r=-1\). Its partial sums alternate between \(4\) and \(0\), so it diverges. The expression \(4/[1-(-1)]\) is invalid because \(|r|<1\) is not satisfied.

Example 4: Terms grow in magnitude

For \(\sum_{n=0}^{\infty}2(3/2)^n\), the ratio is \(3/2\). The terms do not approach zero, so the series diverges.

Example 5: A finite geometric sum

Evaluate \(\sum_{n=0}^{7}2(1/3)^n\). There are eight terms:

No convergence condition is needed for a finite sum.

Example 6: Sum only a tail

For \(\sum_{n=5}^{\infty}7(0.2)^n\), the first included term is \(7(0.2)^5=7/3125\). Thus

Example 7: Convert a repeating decimal

The decimal \(0.\overline{27}\) can be written

Here \(a=27/100\) and \(r=1/100\), so

Example 8: Total distance of a bouncing ball

A ball is dropped \(10\) meters and rebounds to \(60\%\) of each previous height. The initial drop occurs once; every rebound height is traveled upward and downward:

Example 9: Repeated contributions

A process contributes \(200\) units initially, then \(40\%\) of the preceding contribution each stage:

The model converges because each new contribution is a fixed fraction less than one in magnitude.

Example 10: Determine a parameter from a sum

Suppose

Under the required condition \(|x-1|<1\),

The result is valid because \(|7/4-1|=3/4<1\).

Common errors

• Using a difference instead of a ratio to identify the series.
• Calling the coefficient outside the power the first term without checking the lower index.
• Using \(a/(1-r)\) when \(|r|\ge1\).
• Replacing a negative ratio by its absolute value in the denominator.
• Using the infinite-sum formula for a finite series.
• Counting \(N\) terms in a sum from exponent \(0\) through exponent \(N\), which has \(N+1\) terms.
• Starting a tail at \(r^{m-1}\) instead of its actual first exponent \(r^m\).
• Double-counting or omitting the initial movement in a repeated-distance model.
• Reindexing the exponent without changing the bounds.
• Solving for a parameter but failing to verify \(|r|<1\).