Tags » Sequences And Series

The Divergence of the Harmonic Series

The fact that the harmonic series, , diverges has been known since the time of Nicole Oresme in the 14th century, but this fact is still somewhat surprising from a numerical standpoint.   309 more words

Sequences And Series

MATHEMATICAL INDUCTION, SEQUENCES AND SERIES

Mathematical induction:
This is a method of “pulling oneself up by one’s bootstraps” and is regarded with
suspicion by non-mathematicians…

Mathematics

Convergence of a Product

Credits again to Alex for this problem.

Problem. Show that

Proof:

It suffices to show that

Observe that for ,

So, it follows that

Taking and summing, we get… 8 more words

Calculus And Analysis

Synthetic Summer Fun

Today, for some summer fun, let’s look at synthetic division a/k/a synthetic substitution. I’ll assume you all know how to do that since it is a pretty common pre-calculus topic and even comes up again in calculus. 432 more words

Derivatives

On the convergence of sums and products

Credit to Alex Gajewski for sharing this problem with me!

Problem. Let be a nonnegative sequence. Then

Proof: () First observe that for nonnegative , , so by fundamental theorem of Calculus and monotonicity of the integral, we have . 70 more words

Calculus And Analysis

Sequences and Series (Type 10 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. 634 more words

AP Calculus Exams