Pre-Calculus

Sequences, Series, and an Introduction to Limits

Pre-Calculus

✓By the end of this lesson students will be able to define and identify arithmetic and geometric sequences and series.
✓By the end of this lesson students will be able to calculate the sum of finite arithmetic and geometric series.
✓By the end of this lesson students will be able to calculate the sum of infinite geometric series, when convergent.
✓By the end of this lesson students will be able to understand the intuitive concept of a limit of a function.
✓By the end of this lesson students will be able to evaluate limits of functions using direct substitution and algebraic techniques.

Key concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'.

a_n = a_1 + (n-1)d

Arithmetic Series (Finite)

An arithmetic series is the sum of the terms of an arithmetic sequence. For a finite series, the sum of the first 'n' terms can be found using the given formulas.

S_n = n/2 * (a_1 + a_n) OR S_n = n/2 * (2a_1 + (n-1)d)

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by 'r'.

a_n = a_1 * r^(n-1)

Geometric Series (Finite)

A finite geometric series is the sum of the terms of a finite geometric sequence. The sum of the first 'n' terms can be found using the given formula, provided the common ratio 'r' is not equal to 1.

S_n = a_1 * (1 - r^n) / (1 - r), where r ≠ 1

Geometric Series (Infinite)

An infinite geometric series is the sum of the terms of an infinite geometric sequence. This sum converges (exists) only if the absolute value of the common ratio 'r' is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges and has no finite sum.

S = a_1 / (1 - r), where |r| < 1

Introduction to Limits (Intuitive Definition)

The limit of a function describes the behavior of the function as its input (x) approaches a particular value. It represents the value that the function 'approaches' or 'gets arbitrarily close to' as x gets closer and closer to a certain number 'c', without necessarily being equal to 'c'. The notation for a limit is lim (x→c) f(x) = L, meaning 'the limit of f(x) as x approaches c is L'.

Evaluating Limits by Direct Substitution

For many well-behaved functions (like polynomials, rational functions, trigonometric functions, etc.), if 'c' is in the domain of the function, the limit as x approaches 'c' can be found by simply substituting 'c' into the function.

lim (x→c) f(x) = f(c)

Evaluating Limits by Algebraic Techniques

If direct substitution results in an indeterminate form (such as 0/0), algebraic manipulation is often required to simplify the expression before evaluating the limit. Common techniques include factoring, rationalizing the numerator or denominator, and simplifying complex fractions.

Key facts to remember

1An arithmetic sequence has a constant common difference (d).
2A geometric sequence has a constant common ratio (r).
3An infinite geometric series converges (has a finite sum) only if the absolute value of its common ratio |r| < 1.
4The limit of a function describes the value the function approaches as the input approaches a specific value, not necessarily the function's value at that point.
5Direct substitution is the first method to try when evaluating limits for continuous functions.
6An indeterminate form (like 0/0) when evaluating a limit suggests that algebraic simplification is required before substitution.

Worked examples

Example 1

Find the sum of the first 25 terms of the arithmetic series: 5, 9, 13, ...

IIdentify the first term (a_1), common difference (d), and number of terms (n).

IIa_1 = 5

IIId = 9 - 5 = 4

IVn = 25

VUse the formula for the sum of an arithmetic series: S_n = n/2 * (2a_1 + (n-1)d)

VIS_25 = 25/2 * (2*5 + (25-1)*4)

VIIS_25 = 25/2 * (10 + 24*4)

VIIIS_25 = 25/2 * (10 + 96)

9S_25 = 25/2 * (106)

10S_25 = 25 * 53

Answer

S_25 = 1325

Example 2

Find the sum of the first 7 terms of the geometric series: 3, 6, 12, ...

IIdentify the first term (a_1), common ratio (r), and number of terms (n).

IIa_1 = 3

IIIr = 6 / 3 = 2

IVn = 7

VUse the formula for the sum of a finite geometric series: S_n = a_1 * (1 - r^n) / (1 - r)

VIS_7 = 3 * (1 - 2^7) / (1 - 2)

VIIS_7 = 3 * (1 - 128) / (-1)

VIIIS_7 = 3 * (-127) / (-1)

9S_7 = 3 * 127

Answer

S_7 = 381

Example 3

Evaluate the limit: lim (x→-2) (x^2 + 5x + 6) / (x + 2)

IAttempt direct substitution:

II(-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0

III-2 + 2 = 0

IVThis results in the indeterminate form 0/0, so algebraic manipulation is needed.

VFactor the numerator:

VIx^2 + 5x + 6 = (x + 2)(x + 3)

VIIRewrite the limit expression:

VIIIlim (x→-2) [(x + 2)(x + 3)] / (x + 2)

9Since x is approaching -2 but not equal to -2, (x + 2) ≠ 0, so we can cancel the (x + 2) terms.

10lim (x→-2) (x + 3)

11Now, use direct substitution:

12-2 + 3

Answer

When evaluating limits, if direct substitution yields an indeterminate form like 0/0, look for ways to simplify the expression algebraically, such as factoring or rationalizing.

Common mistakes

✗Confusing the formulas for arithmetic and geometric sequences/series.
✗Forgetting to check the condition |r| < 1 before attempting to sum an infinite geometric series.
✗Incorrectly simplifying algebraic expressions when evaluating limits, especially with factoring or fractions.
✗Assuming that the limit of a function as x approaches c is always equal to f(c), without considering cases where the function might be undefined or have a hole at c.
✗Not recognizing indeterminate forms (like 0/0) and stopping the limit evaluation prematurely.

Exam tips

★Always identify the type of sequence or series (arithmetic or geometric) before applying any formulas.
★For infinite geometric series problems, explicitly state whether the series converges or diverges based on the common ratio 'r'.
★When evaluating limits, always attempt direct substitution first. If it yields a numerical value, that's your limit. If it yields an indeterminate form, proceed with algebraic manipulation.
★Show all steps for algebraic simplification when evaluating limits to avoid errors and earn partial credit.

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