📚 BC Series Deep-Dive Notes

Limits, L'Hôpital, Squeeze, MBT

Sequence convergence; L'Hôpital; squeeze theorem; the Monotonic Bounded Theorem; the recursive-sequence three-step playbook.

Partial sums, geometric formula, p-series, n-th term, telescoping

Partial-sum definition; geometric series and convergence; p-series rule; the (one-way) n-th term test; telescoping series.

Decision tree + 7 tests

Decision tree plus integral, comparison, limit-comparison, ratio, and root tests — the heart of BC series.

AST, error bound, classification

The Alternating Series Test, the absolute-vs-conditional distinction, and the AST error bound $|S - S_N| \le b_{N+1}$.

Ratio test for R; endpoint analysis

Find R using the ratio test, then test endpoints individually to nail down the interval — closed or open at each side.

Coefficients $c_n = f^{(n)}(a)/n!$

Derive the coefficient formula and write the three classics: $e^x$, $\sin x$, $\cos x$.

5 base series + substitution / diff / int / algebra

Five must-memorize base series; substitution, differentiation, integration, algebra; computing limits with series.

$|R_n| \le \frac{M}{(n+1)!}|x-a|^{n+1}$

The Lagrange remainder formula, the 3-step method, and the inverse "smallest degree" problem.