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The Fall 2025 Featured Problem Series

Our Featured Problem Series has returned, and will run through December 15 . Look for a new problem every Monday , along with the previous week's solution.

Our problems, which are derived from the courses we tutor, cover a wide range of topics, including Combinatorics , Abstract Algebra , and Real Analysis.

Problems and solutions from our summer series can be found in our archive.

Here is your first challenge.

Our Week 1 Problem

We kick the series off with a sophomore-level real analysis problem like one you may encounter in Penn State Math 312. However, we think the solution should be within the reach of a strong calculus II student.

Suppose that $\set{a_n}_{n=1}^\infty$ is a sequence satisfying $a_n\not=0$ for all $n$, and that $\Lim{n}{\infty} a_n=A$ with $0< \abs{A}<\infty$. Prove that the two series \begin{align*} \sum_{n=1}^\infty\abs{a_{n+1}-a_{n}}&&\text{and}&& \sum_{n=1}^\infty\abs{\frac{1}{a_{n+1}}-\frac{1}{a_{n}}} \end{align*} both either converge or diverge.

The Featured Problem Series Archive


Summer 2025

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