Code Scholars Tutoring · AP Calculus BC
15 topics across 15 lessons, with explanations, worked examples, and self-checks that develop complete mathematical reasoning.
Topics 10.1 · BC only
A sequence is a list of values indexed by integers. An infinite series is studied through a different sequence: its partial sums. The series converges when those partial sums approach a finite number. This distinction separates the behavior of individual terms from the behavior of their accumulated total.
Topics 10.2 · BC only
A geometric series has a constant ratio between successive terms. Its finite partial sum can be written explicitly, which explains why an infinite sum exists when the common ratio has magnitude less than one. Alternating signs are allowed; what matters is the ratio's magnitude.
Topics 10.3 · BC only
If the terms of a series do not approach zero, the series cannot converge. This follows from the partial-sum definition and provides a quick way to rule out convergence. It is a divergence test, not a general test that proves convergence.
Topics 10.4 · BC only
The integral test connects a positive series with the accumulated area of a related continuous function. When the function is positive, continuous, and decreasing beyond some starting point, the series and the corresponding improper integral either both converge or both diverge.
Topics 10.5 · BC only
A p-series sums reciprocal powers of the positive integers. Its convergence depends on whether the exponent is strictly greater than one. The boundary case is the harmonic series, whose terms approach zero but whose partial sums do not approach a finite number.
Topics 10.6 · BC only
Comparison tests use a series whose behavior is already known to study another positive series. Direct comparison needs an inequality in a useful direction. Limit comparison instead shows that two positive term formulas have the same eventual scale.
Topics 10.7 · BC only
An alternating series can converge through cancellation even when the series of term magnitudes diverges. The alternating-series test supplies a sufficient condition: the positive magnitudes decrease eventually and approach zero.
Topics 10.8 · BC only
The ratio test compares the magnitudes of consecutive terms. It is especially useful for factorials and exponential expressions because factors cancel when the ratio is simplified. A limiting ratio below one forces sufficiently rapid decay for absolute convergence.
Topics 10.9 · BC only
Absolute convergence means the series of term magnitudes converges. It guarantees convergence of the original signed series. Conditional convergence means the original series converges but its absolute-value series diverges. Both parts of that definition need justification.
Topics 10.10 · BC only
For an alternating series whose positive magnitudes decrease to zero, the error after a partial sum is no larger than the magnitude of the first omitted term. The sign of that omitted term also indicates which side of the true sum the partial sum lies on.
Topics 10.11 · BC only
A Taylor polynomial approximates a function near a chosen center by matching its derivatives there. The coefficients include factorial denominators, which ensure that differentiating the polynomial the required number of times reproduces the corresponding derivative value.
Topics 10.12 · BC only
The Lagrange remainder bounds Taylor approximation error using a derivative of one order higher than the polynomial degree. A valid bound controls that derivative throughout the interval between the center and the input being approximated.
Topics 10.13 · BC only
A power series is centered at a particular input value. It converges absolutely inside its radius of convergence and diverges outside that radius. The two boundary inputs require separate tests; the radius alone does not say whether either endpoint is included.
Topics 10.14 · BC only
A Taylor series extends the derivative-matching pattern of a Taylor polynomial indefinitely. A Maclaurin series uses center zero. To represent the original function, its remainder must approach zero on the region under discussion; the existence of derivatives alone is not enough.
Topics 10.15 · BC only
A familiar power series can generate new ones through substitution, multiplication, differentiation, or integration. These operations let you represent functions without recomputing every derivative at a center. Track both the algebra and the region where the series representation is justified.
