The Lagrange Highway

Recently, there was an interesting discussion on the AP Calculus Community discussion boards about the Lagrange error bound. You may link to it by clicking here. The replies by James L. Hartman and Daniel J. Teague were particularly enlightening and included files that you may download with the proof of Taylor’s Theorem (Hartman) and its…

Sequences and Series

AP Type Question 10 Sequences and Series – for BC only Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section students may be asked to say if a sequence or series converges or which of several series converge. The Ratio test is used most often to determine…

Error Bounds

How Good is Your Approximation? Whenever you approximate something you should be concerned about how good your approximation is. The error, E, of any approximation is defined to be the absolute value of the difference between the actual value and the approximation. If Tn(x) is the Taylor/Maclaurin approximation of degree n for a function f(x)…

New Series from Old 3

Rational Functions and a “mistake” A geometric series is one in which each term is found by multiplying the preceding term by the same number or expression. This number is called the common ratio, r. Geometric series converge if, and only if, . If a geometric series converges, then the sum of the (infinite number…

New Series from Old 2

Differentiating and integrating a known series can help you find other series. Since  we can find the series for cos(x) this way  Of course we could also have integrated the series for sin(x) to get the series for  –cos(x) and then changed the signs. In out next post we will find that   Recall that…

New Series from Old 1

There are three common ways to get new series from old series without calculating all the derivatives and substituting into the Taylor Series general term. Substituting into known series Differentiating and integrating Approaching rational functions as geometric series This will be  the subject of this and my next two posts. Substituting As you might expect…

Introducing Power Series 3

In my two posts immediately preceding this one I suggested an approach to introducing power series by kind of sneaking up on them starting with the tangent line (local linear) approximation and then going for a second, third and higher degree p[polynomial that had the same value and same derivative values as the function at a point. These…