Sequences and Series: Chapter Review

Vocabulary and notation

Sequence, term $a_n$, index $n$

A function on the positive integers (or on the integers upward from a stated start such as $0$), so $a_n$ is $a(n)$. The index says WHICH term; the term is its value. The graph is isolated dots, never a curve.

Explicit rule and recursive rule

Explicit gives $a_n$ as an expression in $n$, reaching any term in one substitution. Recursive gives seeds plus a step rule and must be climbed rung by rung: a rule looking back $k$ steps needs $k$ seeds, and without them defines nothing.

Sigma notation $\displaystyle\sum_{k=m}^{p} a_k$

Add the summand $a_k$ at every integer from $m$ up to $p$, BOTH limits included. The index is a dummy: renaming it changes nothing and it never appears in the answer. It fixes how many terms there are even when the summand hides it, so $\sum_{k=1}^{6} 5 = 30$.

Series, partial sum $S_n$, converge and diverge

A series adds the terms of a sequence, and $S_n$ adds the first $n$. An infinite series converges when its partial sums close in on a single number $S$, and its sum is DEFINED to be that $S$; otherwise it diverges, with no sum.

Arithmetic sequence, common difference $d$

$a_{n+1} - a_n = d$ for EVERY neighbouring pair, so two matching differences prove nothing. Equivalently $a_n$ is linear in $n$ with slope $d$, so $a_k = 4k + 1$ has $d = 4$ on sight.

Geometric sequence, common ratio $r$

$a_{n+1}/a_n = r$ for every pair, with $a_1 \neq 0$ and $r \neq 0$. Sizes grow when $|r| > 1$ and shrink when $|r| < 1$; a negative $r$ alternates the signs.

Binomial coefficient $\binom{n}{k}$, Pascal's triangle

$\binom{n}{k}$ counts which $k$ of the $n$ factors of $(x+y)^n$ donate their $y$. Row $n$ lists $\binom{n}{0}$ through $\binom{n}{n}$: $n + 1$ entries, second entry $n$.

Factorial $m!$

The product $m(m-1)\cdots 2 \cdot 1$, so $4! = 24$, with $0! = 1$.

Formulas and theorems

• Arithmetic: the $n$th term

  $$\begin{gathered} a_n = a_1 + (n-1)d \\ a_m = a_k + (m-k)d \end{gathered}$$

  Use when Any arithmetic sequence, any $d$ including $0$ and negatives. The multiplier counts STEPS, never terms: $a_1 = 5$ and $d = 3$ give $a_{12} = 5 + 11(3) = 38$. Rearranged, $d = \dfrac{a_m - a_k}{m - k}$ with $m \neq k$.

• Counting terms

  $$\begin{gathered} n = \frac{L - a_1}{d} + 1 \\ m, m+1, \ldots, p \\ \text{has } p - m + 1 \text{ terms} \end{gathered}$$

  Use when The first needs an arithmetic sequence with last term $L$ and $d \neq 0$; at $d = 0$ the values cannot reveal the count. The second holds for any integers $m \le p$, and sigma limits include both ends.

  e.g. $6, 13, \ldots, 97$ has $\tfrac{91}{7} + 1 = 14$ terms.

• Arithmetic series

  $$\begin{gathered} S_n = \frac{n}{2}\left(a_1 + a_n\right) \\ S_n = \frac{n}{2}\bigl(2a_1 + (n-1)d\bigr) \end{gathered}$$

  Use when Constant difference required, and $n$ is the NUMBER of terms, not the last term. Use the first form when the last term is given, the second otherwise.

• The running total of an arithmetic series

  $$\begin{gathered} S_n = \frac{d}{2}n^2 + \left(a_1 - \frac{d}{2}\right)n \\ a_n = S_n - S_{n-1} \end{gathered}$$

  Use when Runs BOTH ways: an arithmetic total is quadratic in $n$ with no constant term, and conversely $S_n = An^2 + Bn$ forces $a_n = 2An + (B - A)$, so $d = 2A$. A nonzero constant term means not arithmetic.

  e.g. $S_n = 3n^2 - n$ gives $a_n = 6n - 4$ and $d = 6$.

• Geometric: the $n$th term

  $$a_n = a_1 r^{\,n-1}, \qquad \frac{a_m}{a_k} = r^{\,m-k}$$

  Use when $a_1 \neq 0$ and $r \neq 0$. The exponent is $n-1$, one per multiplication between term $1$ and term $n$. Keep a negative ratio bracketed: $a_1 = 3$ and $r = -2$ give $a_6 = 3(-2)^5 = -96$.

• Finite geometric series

  $$\begin{gathered} S_n = \frac{a\left(1 - r^{\,n}\right)}{1 - r} = \frac{a\left(r^{\,n} - 1\right)}{r - 1} \quad (r \neq 1) \\ S_n = na \quad (r = 1) \end{gathered}$$

  Use when The fraction needs $r \neq 1$: the denominator $1 - r$ vanishes there; at $r = 1$ every term equals $a$, so the sum is $na$. Only $r = 1$ is excluded: $r = -1$ gives a denominator of $2$.

  e.g. $4 - 8 + 16 - 32 + 64 = \dfrac{4\left(1 - (-2)^5\right)}{1 - (-2)} = \dfrac{4(33)}{3} = 44$.

• Infinite geometric series

  $$\begin{gathered} \sum_{n=1}^{\infty} a r^{\,n-1} = \frac{a}{1 - r} \\ S - S_n = S\,r^{\,n} \end{gathered}$$

  Use when Needs $|r| < 1$, a condition on SIZE, so it covers negative ratios. With $a \neq 0$ and $|r| \ge 1$ the partial sums never settle and there is no sum. $S - S_n$ is the exact error left after $n$ terms.

  e.g. $6 - 2 + \tfrac23 - \cdots = \dfrac{6}{1 - \left(-\tfrac13\right)} = \dfrac{6}{\tfrac43} = \dfrac92$.

• The Binomial Theorem

  $$\begin{gathered} (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{\,n-k} y^{\,k} \\ (x-y)^n = \sum_{k=0}^{n} (-1)^k \binom{n}{k} x^{\,n-k} y^{\,k} \end{gathered}$$

  Use when $n$ a whole number. Exactly $n + 1$ terms, and every term's two exponents sum to $n$. For a difference the coefficient is negative exactly when $k$ is odd.

• General term of an expansion

  $$\begin{gathered} T_k = \binom{n}{k} A^{\,n-k} B^{\,k} \\ k = 0, 1, \ldots, n \end{gathered}$$

  Use when $A$ and $B$ are the WHOLE terms of $(A+B)^n$, signs included. The list starts at $k = 0$, so $T_k$ is the $(k+1)$st term and the $r$th term is $\binom{n}{r-1}A^{\,n-r+1}B^{\,r-1}$.

• Binomial coefficients

  $$\begin{gathered} \binom{n}{k} = \frac{n!}{k!\,(n-k)!} \\ \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} \\ \binom{n}{k} = \binom{n}{n-k} \end{gathered}$$

  Text description

  Rows 0 to 5 of Pascal's triangle, with the neighbouring entries 4 and 6 joined by short lines down to the 10 they sum to in the row below.

  Use when Integers with $0 \le k \le n$, and $0! = 1$. Pascal's rule needs an interior $1 \le k \le n-1$; the ends are held by $\binom{n}{0} = \binom{n}{n} = 1$. Row $n$ totals $2^n$. Cancel $(n-k)!$ before multiplying.

  e.g. $\binom{10}{3} = \dfrac{10 \times 9 \times 8}{3!} = 120$.

Problem types, step by step

Exam traps

• Trap Multiplying the step by the term number: for $4, 11, 18, \ldots$, writing $a_{30} = 4 + 30(7) = 214$.

  Fix $214$ is the 31st term: reaching term $n$ takes $n-1$ steps, so $a_{30} = 4 + 29(7) = 207$. The exponent in $a_1r^{\,n-1}$ counts the same way, so the 5th term of $2, 6, 18, \ldots$ is $2 \cdot 3^4 = 162$. Recovering $d$ or $r$ from two terms divides by steps too.

• Trap Losing the $+1$ that turns steps back into terms.

  Fix $(L - a_1) \div d$ counts steps: from $3$ to $43$ by fours is $10$ steps but $11$ terms. Sigma limits are inclusive too, so $\sum_{k=3}^{6}$ holds $4$.

• Trap Reading 'diverges' as 'blows up', then trusting $\dfrac{a}{1-r}$ at $r = -1$.

  Fix $7 - 7 + 7 - 7 + \cdots$ has bounded partial sums $7, 0, 7, 0, \ldots$ that never settle, so it has no sum, though the formula would report $\tfrac72$.

• Trap Sign slips with a negative ratio.

  Fix $1 - \left(-\tfrac13\right) = \tfrac43$, never $\tfrac23$: subtracting a negative adds. An even exponent kills the sign too, so $(-3)^6 = +729$.

• Trap Calling $T_k$ the $k$th term of an expansion.

  Fix The list starts at $k = 0$, so $T_k$ is the $(k+1)$st term. The 3rd term of $(x+y)^7$ has $k = 2$, giving $21x^5y^2$, not the $35x^4y^3$ that $k = 3$ returns.

• Trap Leaving a coefficient outside the power: reading the $k = 1$ term of $(2a+3)^4$ as $4a^3(3) = 12a^3$.

  Fix The whole bracketed quantity is raised, so that term is $4(2a)^3(3) = 96a^3$.

• Trap Marking every coefficient of $(x - y)^n$ negative because the binomial has a minus sign.

  Fix The sign is $(-1)^k$, so it is negative exactly when $k$ is odd. In $(x-1)^6$ the coefficient of $x^4$ is $+15$.

• Trap Reading a percentage change as the common ratio.

  Fix A rise of $6$ percent gives $r = 1.06$, not $0.06$; a drop of $20$ percent gives $r = 0.80$.

• Trap Averaging to find a term sitting between two others in a geometric sequence.

  Fix The ratio must stay constant, so the term between $4$ and $25$ satisfies $x^2 = 100$, giving $x = 10$ or $x = -10$, not $14.5$.

• Trap Assuming shrinking terms force convergence, so any series whose terms go to $0$ has a sum.

  Fix That is a geometric fact, not a general one: $1 + \tfrac12 + \tfrac13 + \cdots$ has terms shrinking to $0$ and partial sums past every bound. For a geometric series the test is $\lvert r \rvert < 1$.