MZB221 Advanced Engineering Mathematics - Week 1: Infinite Series

Vocabulary based on the introduction and core mathematical definitions for Week 1 of MZB221 Advanced Engineering Mathematics.

Sequence Limit

A sequence ${a_n}$ has the limit $L$ if $a_n$ approaches $L$ as $n \rightarrow \infty$.

Partial Sum ($S_n$)

Defined as $S_n = \sum_{j=1}^n a_j$, it is a new sequence that arises as the sum of terms in another sequence ${a_n}$.

Geometric Series Convergence

The series $\sum_{n=1}^\infty r^n$ converges to $\frac{1}{1-r}$ if $|r| < 1$ and diverges otherwise.

Sum of a Geometric Series ($\sum_{n=0}^\infty ar^n$)

For $|r| < 1$, the sum is given by $\frac{a}{1-r}$.

Harmonic Series

The infinite series $\sum_{n=1}^\infty \frac{1}{n}$, which is known to diverge.

Ratio Test

A test where the series $\sum_{n=1}^\infty a_n$ converges if $\rho = \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$ and diverges if $\rho > 1$.

Alternating Series Test

A series $\sum_{n=1}^\infty (-1)^{n-1} a_n$ converges if $a_n$ is positive and decreasing and $\lim_{n \rightarrow \infty} a_n = 0$.