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

## Tags » Sequences And Series

#### 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

#### 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

#### 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

#### 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