Glossary
Alternating Harmonic Series
A series similar to the harmonic series but with alternating signs, typically represented as $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$.
Example:
Unlike the regular harmonic series, the alternating harmonic series actually converges to a finite value.
Converges
A series *converges* if its sequence of partial sums approaches a finite, specific limit. This means the sum of its terms adds up to a definite number.
Example:
If you keep adding terms of , the sum will eventually approach a finite value, meaning the series converges.
Diverges
A series *diverges* if its sequence of partial sums does not approach a finite limit, meaning the sum grows infinitely large, infinitely small, or oscillates without settling.
Example:
Even though the terms get smaller, the harmonic series diverges because its sum continues to grow without bound.
Harmonic Series
A specific type of p-series where the exponent p equals 1, represented as the sum of the reciprocals of the positive integers.
Example:
The series 1 + 1/2 + 1/3 + 1/4 + ... is a classic example of a harmonic series and is known to diverge.
Summation Notation
A concise mathematical notation using the Greek capital letter sigma ($\Sigma$) to represent the sum of a sequence of terms.
Example:
The expression uses summation notation to represent the sum .
p-series
A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$, where p is a positive real number. Its convergence depends on the value of p.
Example:
The series is a p-series with p=2, which converges because p > 1.