Topic 6 of 6
Sequences & Series
Decide convergence, work with power series, and understand what finite approximations can guarantee.
01Direct answerWhy does the harmonic series diverge?Its terms approach zero, but they do so too slowly for the accumulated sum to settle.02Concept explainerGeometric series: convergence, sum, and structureA constant ratio makes an infinite sum exactly solvable when repeated scaling shrinks toward zero.03Decision guideHow to choose a convergence testMatch the series’ structure to a test instead of cycling through tests in chapter order.04Method guideFinding a power series interval of convergenceFind the radius with the Ratio or Root Test, then test both endpoints separately because the general test goes silent there.05Concept explainerTaylor remainder: how accurate is the polynomial?A Taylor polynomial is useful only with a way to control what was left out.