AP Calculus BC

Lagrange vs Alternating Series Error Bound

The alternating series error bound is the absolute value of the first omitted term, and it applies only when the series alternates and passes the alternating series test. The Lagrange error bound works for any Taylor polynomial but requires bounding the next derivative on the interval.

Alternating series bound

Use when: The series alternates and satisfies the alternating series test, which makes the bound a single term.

Lagrange error bound

Use when: The series does not alternate, or the question explicitly asks for the Lagrange bound.

Side by side

	Alternating bound	Lagrange bound
Formula	$\lvert a_{N+1} \rvert$	$\frac{\max\lvert f^{(n+1)} \rvert}{(n+1)!}\lvert x-a \rvert^{n+1}$
Requires	An alternating series	A bound on the next derivative
Effort	One term	Finding a maximum
Tightness	Usually tight	Often loose

The alternating bound is simple because the partial sums overshoot and undershoot the true sum in turn, so the sum is always trapped between consecutive partial sums. That structure does not exist for a general series, which is why Lagrange needs more work.

For sine and cosine the Lagrange bound simplifies a lot, since every derivative is bounded by one and the maximum is simply one.

The alternating trap

Use the first term you dropped, not the last one you kept. The bound is the size of the next term, the one not in your partial sum.

Frequently asked questions

Is the Lagrange bound the actual error?

No, it is an upper bound and is often much larger than the true error. Questions ask you to show the error is under some amount, which the bound does even when loose.

In the CED: Unit 10: Infinite Sequences and Series (BC)