Telescoping Series

Algebra Progress

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Telescoping Series

An introduction to the telescoping series and its appearances in contests.

Authors: Jaden Park, Leo Zhang

Formula

Telescoping series usually appear on harder math contests. They are in the form of $t_{n} = \frac{n}{n + 1}.$

Practice Problem

Evaluate the sum of

\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \ldots + \frac{1}{99 \cdot 100}.

Here are some practice problems.

Pratice Problem

(2004 Purple Comet HS, Q21)

Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_4 + \ldots + a_k.$ Let $\frac{a_{100}}{b_{100}} = \frac{m}{n}$ where $m$ and $n$ are relatively prime natural integers. Find $n - m.$

Pratice Problem

(February 2006 HMMT General Part 2, Q2)

Note: Telescoping expressions are not just limited to sums. Nasty products as well can be significantly simplified through rampant cancellation.

Find

\left( \frac{2^2}{2^2 - 1} \right) \cdot \left( \frac{3^2}{3^2 - 1} \right) \cdot \left( \frac{4^2}{4^2 - 1} \right) \ldots \left( \frac{2006^2}{2006^2 - 1} \right)

Final Notes and Tips

Pretty common to see, would recommend memorizing the pattern

Arithmetico-Geometric Series

Recursion

Algebra Progress

Polynomials

Sequence and Series

Miscellaneous

Formula

Practice Problem

Pratice Problem

Pratice Problem

Final Notes and Tips

Formula

Practice Problem

Pratice Problem

Pratice Problem

Final Notes and Tips