Series flashcards

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What is the difference between a sequence and a series?

A sequence is a list of numbers in a specific order (e.g., 2, 4, 6, 8). A series is the sum of the terms of a sequence (e.g., 2 + 4 + 6 + 8).

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What is the formula for the n-th term of an Arithmetic Progression (AP)?

$u_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.

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What are the two formulae for the sum of the first n terms of an AP?

$S_n = \frac{n}{2}(2a + (n-1)d)$ or $S_n = \frac{n}{2}(a+l)$, where $l$ is the last term.

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What is the formula for the n-th term of a Geometric Progression (GP)?

$u_n = ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

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What is the formula for the sum of the first n terms of a GP?

$S_n = \frac{a(1-r^n)}{1-r}$ or $S_n = \frac{a(r^n-1)}{r-1}$, where $r \neq 1$.

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What is the condition for a geometric series to converge (have a finite sum to infinity)?

The common ratio $r$ must satisfy $|r| < 1$, which is the same as $-1 < r < 1$.

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What is the formula for the sum to infinity of a convergent geometric series?

$S_\infty = \frac{a}{1-r}$, valid only for $|r| < 1$.

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What is the binomial expansion for $(1+x)^n$ where $n$ is any rational number?

$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$

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What is the condition on $x$ for the binomial expansion of $(1+x)^n$ to be valid when $n$ is not a positive integer?

The expansion is valid for $|x| < 1$.

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Common Trap: What must you do before expanding $(a+bx)^n$ using the standard binomial formula?

You must first factor out $a$ to get it into the form $a^n(1 + \frac{b}{a}x)^n$. The expansion is then valid for $|\frac{b}{a}x| < 1$.

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Common Trap: A question asks for the set of values for which a GP converges. What is the answer?

You must solve the inequality $|r| < 1$ for the variable in the expression for $r$. For example, if $r = (x-1)$, then $|x-1|<1 \implies -1 < x-1 < 1 \implies 0 < x < 2$.

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Arithmetic progression

A sequence in which each term after the first is found by adding a constant, the common difference $d$, to the previous term: $u_n = a + (n-1)d$. For example 5, 8, 11, 14, … has $a = 5$ and $d = 3$.

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Common difference

The constant $d$ added to each term of an arithmetic progression to give the next: $d = u_{n+1} - u_n$, the same for every pair of consecutive terms.

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Geometric progression

A sequence in which each term after the first is found by multiplying the previous term by a constant, the common ratio $r$: $u_n = ar^{n-1}$. For example 3, 6, 12, 24, … has $a = 3$ and $r = 2$.

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Common ratio

The constant $r$ that each term of a geometric progression is multiplied by to give the next: $r = u_{n+1} \div u_n$, the same for every pair of consecutive terms.

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Sum to infinity

The finite value $S_\infty = \dfrac{a}{1-r}$ that the sum of a geometric progression approaches as more and more terms are added, which exists only when $|r| < 1$; if $|r| \ge 1$ the sum does not converge.

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Convergent series

A series whose partial sums approach a finite limit as the number of terms grows. A geometric series converges exactly when its common ratio satisfies $-1 < r < 1$.