Language of Sequences & Series

A series is the mathematical result of adding the terms of a sequence together. While a sequence is a list, a series represents a single cumulative value or a running total of those listed values.

The notation $S_n$ is used to represent the sum of the first $n$ terms of a sequence. For example, $S_3 = u_1 + u_2 + u_3$, representing the total accumulation up to the third position.

The relationship between a sequence and its sum can be expressed as $S_n = \sum_{i=1}^{n} u_i$. This summation notation provides a compact way to describe the addition of many terms following a specific rule.

An increasing sequence is one where every subsequent term is strictly greater than the term preceding it. Mathematically, this is defined by the condition $u_{n+1} > u_n$ for all positive integers $n$.

Conversely, a decreasing sequence occurs when every term is strictly less than the one before it, satisfying the condition $u_{n+1} < u_n$. This indicates a consistent downward trend in the values of the terms.

Some sequences are monotonic, meaning they only move in one direction (either always increasing or always decreasing), while others may fluctuate or remain constant.

A periodic sequence is characterized by a repeating cycle of values. After a certain number of terms, the sequence returns to its starting value and repeats the exact same pattern indefinitely.

The order (or period) of a periodic sequence is the number of terms in one complete repeating cycle. For instance, a sequence that alternates between two values has an order of 2.

Periodic behavior is frequently found in sequences involving trigonometric functions or powers of negative numbers. For example, the sequence defined by $u_n = \sin(90n^\circ)$ repeats every four terms, giving it an order of 4.

It is vital to distinguish between the position ($n$) and the value ($u_n$). The position $n$ must always be a positive integer ($1, 2, 3, \dots$), whereas the value $u_n$ can be any real number, including fractions, decimals, or negatives.

The relationship $u_n = S_n - S_{n-1}$ is a critical tool for finding a specific term when only the sum formula is known. This works because subtracting the sum of $(n-1)$ terms from the sum of $n$ terms leaves only the final term.

• Always check $n=1$: When given a general formula for $u_n$ or $S_n$, substitute $n=1$ first. For any sequence, $u_1$ must equal $S_1$, which serves as an excellent verification step.

• Identify Periodic Patterns: If a sequence involves $(-1)^n$ or trigonometric functions like $\sin$ or $\cos$, immediately test the first few terms to see if they repeat. This often reveals a periodic nature that simplifies complex summation problems.

• Verify Monotonicity: To prove a sequence is increasing or decreasing, examine the sign of the difference $u_{n+1} - u_n$. If the difference is always positive, the sequence is increasing; if always negative, it is decreasing.

• Watch the Index: Ensure you are using the correct bounds for $n$. In most academic contexts, sequences start at $n=1$, but always verify if a specific problem starts at $n=0$ or another value.