Sequences and Series of Real Numbers — Convergence

GRE Math Subject · Axiom Academy

LESSON Sequences & Series — Convergence Cauchy sequences, Bolzano–Weierstrass, and every convergence test the GRE demands Step 1: Convergence of Sequences Definition: A sequence in converges to if: For every , there exists such that . Key Properties of Convergent Sequences: Every convergent sequence is bounded (but the converse is false — consider ) Limits are unique : if and , then Algebraic limit theorem: Limits respect addition, scalar multiplication, products, and quotients (when denominator limit is nonzero) Every monotone bounded sequence converges (Monotone Convergence Theorem) GRE Quick Hit: Does converge? Yes — it is monotone decreasing and bounded below by 0. Its limit is . Step 2: Cauchy Sequences and Completeness Definition: A sequence is Cauchy if: Theorem (Completeness of R): A sequence of real numbers converges if and only if it is Cauchy. This is what makes complete . The rationals are not complete — for instance, the decimal approximations form a Cauchy sequence in that does not converge in . Bolzano–Weierstrass Theorem: Every bounded sequence of real numbers has a convergent subsequence. Equivalently: every bounded infinite subset of has a limit point. This is one of the most frequently tested theorems on the GRE. Step 3: Series — Definitions and Partial Sums Definition: The series converges if the sequence of partial sums converges. Divergence Test (nth-term test): If , then diverges. Caution: The converse is false! has , yet the harmonic series diverges.

This is the written version of the interactive lesson above. See the full GRE Math Subject course.