Why One Series Explodes While Another Shrinks

Display the two sums side by side and invite comparison: harmonic terms decrease gently, while powers-of-two terms quickly become extremely small.

• Both sequences of positive terms approach zero.
• Harmonic: 1, 1/2, 1/3, 1/4, …
• Geometric: 1/2, 1/4, 1/8, 1/16, …
• The puzzle asks why their infinite totals behave differently

Step through both sums in lockstep. Before the evidence, choose which sum will eventually exceed 2.

• Harmonic partial sum: H_n = 1 + 1/2 + … + 1/n
• Geometric partial sum: G_n = 1/2 + 1/4 + … + 1/2^n
• Commit to which partial-sum curve has no finite ceiling
• Do not reveal the result before interacting

Add powers-of-two index blocks one at a time. Reveal each block's lower bound to test whether enough total mass remains to cross the next integer.

• Block 1: terms with n from 2 through 2 each satisfy 1/n ≥ 1/2
• Block 2: terms with n from 3 through 4 each satisfy 1/n ≥ 1/4
• Later block k contains 2^(k-1) terms, each at least 1/2^k
• Every block contributes at least 1/2, so selected block sums accumulate without bound

Use the block evidence to explain the divergence mechanism: terms approach zero, but the number of terms in each block grows fast enough that its sum does not vanish.

• The block from 2^(k-1)+1 through 2^k has 2^(k-1) terms
• Each term is at least 1/2^k
• Block sum is at least 2^(k-1)/2^k = 1/2
• Infinitely many blocks therefore produce an unbounded partial-sum sequence

Build a unit-length bar, remove its first half, then halve the remainder repeatedly. Compare the visible subtractive pattern with an additive rearrangement.

• The bar starts with length 1
• Subtract 1/2, then 1/4, then 1/8, and so on
• The removed lengths are exactly the series terms
• No positive interval remains after infinitely many halvings

Distinguish the essential mechanisms from the mistaken claim that merely having terms approach zero guarantees a finite sum.

• Terms approaching zero is necessary for convergence of a positive series, not sufficient
• The harmonic blocks contribute at least 1/2 each
• The geometric remainder becomes 1/2, 1/4, 1/8, … and tends to zero
• Changing the geometric ratio to 1 would produce divergence

Apply the comparison to 1 + 1/2 + 1/3 + … versus 1 + 1/3 + 1/9 + … using exactly one focused choice.

• Use block mass or remaining total rather than term size alone
• Select the comparison that best predicts each outcome

Answer the driving question directly by placing the two mechanisms beside one another and explicitly evaluate the geometric series.

• Harmonic: after the initial 1, every selected dyadic block adds at least 1/2, so partial sums exceed every finite bound
• Geometric: 1/2 + 1/4 + 1/8 + … = 1/2 + 1/2 + 1/2 + … because the tail after each term is again half that term
• The n-term geometric partial sum is 1 - 1/2^n
• Therefore the geometric sum is exactly 1, while the harmonic series diverges