Harmonic Series Divergence

Calculus 2 · Axiom Academy

LESSON Harmonic Series Divergence How an infinite sum of terms shrinking to zero can still pile up to infinity The harmonic series adds the reciprocals of every counting number. Each term is smaller than the one before, and the terms march down toward zero. Stack those same bars end to end and you get the partial sum — the running total. It rises slowly, but it never levels off: it sails past every horizontal line you draw. The growth is genuinely slow — but unstoppable. Look how far you have to go just to add the next to the total: Here is the classic proof. Bracket the terms into blocks whose lengths double — , , , , and so on. Replace every term in a block by its smallest member and each block collapses to exactly . Key insight. Each block of 2^ k terms sums to at least , because it holds 2^ k terms that are each , and . With infinitely many blocks, the total is at least , which has no ceiling. The harmonic series teaches a crucial fact about every infinite series. Watch two series whose terms both shrink to zero — yet one converges and one does not. For convergence you must have . If the terms do not approach 0 , the series definitely diverges. Even with , the series might still diverge — exactly what the harmonic series proves. How fast the terms shrink decides everything. For : it converges if and diverges if . p=1 is the harmonic series — the dividing line. It is the canonical " but diverges" example.

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