Harmonic Series

Algebra Progress

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Harmonic Series

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

Author: Jaden Park

Formula

A Harmonic sequence is a sequence of the form

a_n = \left(\frac{1}{a_1 + (n - 1)d} \right)

\left(\frac{1}{a} \right), \left(\frac{1}{a + d}\right), \left(\frac{1}{a + 2d}\right), \ldots, \left(\frac{1}{a + (n - 1)d}\right).

In other words, the reciprocals of an arithmetic sequence form a harmonic sequence.

The sum of the first $n$ terms of a harmonic sequence is

S_n = \left(\frac{n}{a_1 + (n - 1)d} \right).

(It's not neccessary to go over the formula for the sum because contests only go over the sequence, not the series.)

Here are some practice problems.

Practice Problem

If the sum of the first $2$ terms of a harmonic sequence is $\frac{17}{70},$ the sum of the next $2$ terms is $frac{5}{4},$ and the sum of the following $2$ terms if $-\frac{7}{10},$ find the sum of the following $2$ terms.

Practice Problem

Let $a$ , $b$ , and $c$ be positive real numbers. Show that if $a$ , $b$ , and $c$ are in harmonic progression, then $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ are as well.

Practice Problem

(1959 AHSME Q13)

A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let $S_n$ represent the sum of the first $n$ terms of the harmonic progression; for example $S_3$ represents the sum of the first three terms. If the first three terms of a harmonic progression are $3, 4, 6$ , then:

Final Notes and Tips

Not super common to see, but would recommend memorizing what it looks like in general.

Geometric Series

Arithmetico-Geometric Series

Algebra Progress

Polynomials

Sequence and Series

Miscellaneous

Formula

Practice Problem

Practice Problem

Practice Problem

Final Notes and Tips

Formula

Practice Problem

Practice Problem

Practice Problem

Final Notes and Tips