Sequences, Series and Sigma Notation

A sequence is an ordered list of objects, usually numbers, where each element occupies a specific position. The position number $n$ is always a positive integer.

Notation and term rules:

• $T_n$ notation: $T_1$ is the first term, $T_2$ the second, $T_n$ the $n$th term. The subscript is the position.
• Explicit (general term) formula: gives $T_n$ directly in terms of $n$. For $T_n = 2n$: $T_1 = 2$, $T_2 = 4$, $T_3 = 6$, $\ldots$ (every even number). For $T_n = n^2$: $T_1=1$, $T_2=4$, $T_3=9$, $\ldots$ (perfect squares).
• Recurrence (recursive) relation: defines each term using the previous one. $T_1 = 3$, $T_n = T_{n-1} + 4$ gives: $3, 7, 11, 15, \ldots$
• Finite sequence: terminates, written without ellipsis, e.g. $2, 4, 6, 8, 10$. Has $N$ terms.
• Infinite sequence: continues without end, written with ellipsis, e.g. $2, 4, 6, 8, 10, \ldots$

Sequence = ordered list of terms. $T_n$ = $n$th term ($n$ is a positive integer).; Explicit formula: $T_n = f(n)$ gives the term directly. Recurrence: $T_n = g(T_{n-1})$ defines each term from the previous one.

Pause, copy the definitions: sequence = ordered list of terms; $T_n$ = $n$th term; explicit formula gives $T_n$ directly from $n$; recurrence defines $T_n$ from the previous term $T_{n-1}$, into your book.

We just saw that a sequence is an ordered list of terms described by an explicit formula $T_n = f(n)$ or a recurrence relation. That raises a question: given the terms, how do we compute their running total, and can we recover any individual term from the total? This card answers it → the partial sum $S_n = T_1 + \cdots + T_n$ is itself a new sequence, and the identity $T_n = S_n - S_{n-1}$ recovers any term from consecutive partial sums.

The $n$th partial sum $S_n$ is the sum of the first $n$ terms of a sequence. Partial sums themselves form a new sequence, and understanding their relationship to $T_n$ is critical for HSC problems.

Computing partial sums, example with $T_n = 2n$:

• $S_1 = T_1 = 2$
• $S_2 = T_1 + T_2 = 2 + 4 = 6$
• $S_3 = T_1 + T_2 + T_3 = 2 + 4 + 6 = 12$
• $S_4 = 2 + 4 + 6 + 8 = 20$
• $S_5 = 2 + 4 + 6 + 8 + 10 = 30$

The partial sums $2, 6, 12, 20, 30, \ldots$ form their own sequence. Notice the closed-form: $S_n = n(n+1)$ in this case.

$S_n = T_1 + T_2 + \cdots + T_n$, the sum of the first $n$ terms.; $T_n = S_n - S_{n-1}$ for $n \geq 2$. Always check $T_1 = S_1$ separately.

Pause, copy the partial sum definition $S_n = T_1 + T_2 + \cdots + T_n$ and the key identity $T_n = S_n - S_{n-1}$ for $n \geq 2$ (always check $T_1 = S_1$ separately) into your book.

We just saw the partial sum $S_n$ as the cumulative total of the first $n$ terms. That raises a question: what happens when the sequence is infinite, does adding infinitely many terms always blow up, or can the total converge to a finite value? This card answers it → a finite series is just $S_N$; an infinite series is $\lim_{n\to\infty} S_n$, which may converge (like $1 + \frac{1}{2} + \frac{1}{4} + \cdots = 2$) or diverge (like $1 + 2 + 3 + \cdots$).

A series is the sum of the terms of a sequence. When a sequence is finite, the series is simply $S_N$. When the sequence is infinite, the series is a limit, it may or may not settle on a finite value.

Types of series:

• Finite series: $S_N = T_1 + T_2 + \cdots + T_N$, a definite number. The stadium problem is a finite series: $S_{50} = 1 + 2 + \cdots + 50 = 1275$.
• Infinite series: $S = \displaystyle\lim_{n \to \infty} S_n = T_1 + T_2 + T_3 + \cdots$, a limit. This may exist (converge) or not (diverge).
• Diverging series example: $1 + 2 + 3 + \cdots$, partial sums grow without bound. No finite sum exists.
• Converging series example (preview): $1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots$, partial sums approach 2. The infinite sum is exactly 2. (Full convergence theory is Lesson 4.)

Finite series = sum of all terms of a finite sequence = $S_N$.; Infinite series = $\lim_{n\to\infty} S_n$, may converge (finite sum) or diverge (no finite sum).

Pause, copy the distinction: finite series = $S_N$; infinite series = $\lim_{n\to\infty} S_n$, may converge (geometric with $|r| < 1$) or diverge ($1+2+3+\cdots$ has no finite limit), into your book.

We just saw that a series is a sum of terms, finite or infinite. That raises a question: writing out $T_1 + T_2 + \cdots + T_n$ becomes cumbersome for long sums, is there a compact notation that specifies the general term, starting index, and ending index in one expression? This card answers it → sigma notation $\displaystyle\sum_{k=r}^{n} T_k$ compresses the entire sum into a single symbol.

Sigma notation $\displaystyle\sum_{k=r}^{n} T_k$ compresses a sum into a single expression. Reading and writing sigma notation fluently is essential for every HSC series topic.

Anatomy of sigma notation:

• $\Sigma$ (capital sigma): means "sum".
• Lower limit ($k = r$): the starting value of the index. Most commonly $k = 1$.
• Upper limit ($n$): the ending value of the index. Can be a specific number or $\infty$ for an infinite series.
• General term ($T_k$): the expression you substitute $k$ into for each step.

Examples with computed values:

• $\displaystyle\sum_{k=1}^{4}(2k-1) = 1 + 3 + 5 + 7 = 16$ (sum of first 4 odd numbers)
• $\displaystyle\sum_{k=3}^{7}k^2 = 9 + 16 + 25 + 36 + 49 = 135$ (sum starts at $k=3$, not $k=1$)
• $\displaystyle\sum_{k=1}^{10}(2k-1) = $ sum of first 10 odd numbers $= 1+3+5+\cdots+19 = 100$
• $\displaystyle\sum_{k=1}^{\infty}\frac{1}{2^k} = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots = 1$ (converging infinite series)

$\displaystyle\sum_{k=r}^{n} T_k$: sum the general term $T_k$ for $k = r, r+1, \ldots, n$.; Infinite series: upper limit is $\infty$. May converge or diverge.

Pause, copy sigma notation $\displaystyle\sum_{k=r}^{n} T_k$ (lower limit $r$, upper limit $n$, index $k$ is a dummy variable) and the infinite form $\displaystyle\sum_{k=r}^{\infty} T_k$ into your book.

We just saw sigma notation as a compact way to write any sum. That raises a question: the HSC often gives you a formula for $S_n$ and asks you to find $T_n$, how do you extract the general term from a sum formula, and what can go wrong at $n = 1$? This card answers it → use $T_n = S_n - S_{n-1}$ for $n \geq 2$, then verify $T_1 = S_1$; if the check fails, state $T_1$ as a separate piecewise value.

A common HSC question gives you a formula for $S_n$ (the partial sum) and asks you to find $T_n$ (the general term). The technique is $T_n = S_n - S_{n-1}$, with a separate check for $T_1$.

Worked example: $S_n = n^2 + 3n$

• Step 1, find $T_n$ for $n \geq 2$: $$T_n = S_n - S_{n-1} = (n^2 + 3n) - \bigl((n-1)^2 + 3(n-1)\bigr)$$ $$= n^2+3n - (n^2-2n+1+3n-3) = n^2+3n - n^2 - n + 2 = 2n+2$$
• Step 2, check $T_1$ using $S_1$: $S_1 = 1^2 + 3(1) = 4$. Does the formula give $T_1 = 2(1)+2 = 4$? Yes ✓. So $T_n = 2n+2$ works for all $n \geq 1$.
• Step 3, find specific terms: $T_5 = 2(5)+2 = 12$. Verify: $S_5 = 25+15 = 40$ and $S_4 = 16+12 = 28$. $S_5 - S_4 = 12$ ✓.

Given $S_n$: use $T_n = S_n - S_{n-1}$ for $n \geq 2$ to find the general term.; Always check: does $T_1 = S_1$? If yes, formula holds for all $n$. If no, state $T_1$ separately.

Pause, copy the technique: $T_n = S_n - S_{n-1}$ for $n \geq 2$; always verify $T_1 = S_1$; if the check fails, state $T_1$ separately as a piecewise formula, into your book.