Sequences, Series, and Limits

Grade 11 · Math · Study Guide

A comprehensive overview of arithmetic and geometric sequences, series notation, and the concept of limits.

A sequence is an ordered list of numbers following a specific rule. Arithmetic sequences have a constant difference between consecutive terms, while geometric sequences have a constant ratio. The nth term of an arithmetic sequence is given by a_n = a_1 + (n-1)d, where d is the common difference. For geometric sequences, a_n = a_1 * r^(n-1), where r is the common ratio.

A series is the sum of terms in a sequence. The sum of the first n terms of an arithmetic series is S_n = n/2 * (a_1 + a_n), and for a geometric series S_n = a_1(1 - r^n)/(1 - r) when r ≠ 1. An infinite geometric series converges when |r| < 1, and its sum is S = a_1/(1 - r).

Limits describe the value a function or sequence approaches as the input or index grows without bound. The notation lim(n→∞) a_n = L means the terms get arbitrarily close to L. A sequence is convergent if its limit exists and divergent otherwise. Understanding limits is the foundation of calculus and is essential for evaluating infinite series and understanding continuity.

Key terms

Arithmetic Sequence: A sequence where the difference between consecutive terms is constant, called the common difference d.
Geometric Sequence: A sequence where the ratio between consecutive terms is constant, called the common ratio r.
Series: The sum of the terms of a sequence, often expressed using sigma notation Σ.
Convergence: A sequence or series converges if it approaches a finite limit as the number of terms increases without bound.
Partial Sum: The sum S_n of the first n terms of a series.
Limit: The value that a sequence or function approaches as its index or input approaches a specified value or infinity.

Flashcards

Arithmetic sequence nth term: a_n = a_1 + (n - 1)d
Geometric sequence nth term: a_n = a_1 * r^(n - 1)
Sum of arithmetic series: S_n = n/2 * (a_1 + a_n)
Sum of finite geometric series: S_n = a_1(1 - r^n) / (1 - r), r ≠ 1
Sum of infinite geometric series: S = a_1 / (1 - r), valid only when |r| < 1
Divergent sequence: A sequence whose terms do not approach any finite limit as n → ∞
Sigma notation: Σ is used to represent the sum of a series; Σ_{k=1}^{n} a_k means sum from k=1 to n
Limit of a sequence: lim(n→∞) a_n = L means the sequence terms get arbitrarily close to L

Questions

What is the sum of the infinite geometric series 3 + 1 + 1/3 + 1/9 + ...?
1. 4
2. 4.5
3. 9/2
4. 3/2
Show answer
Answer: 4.5
First term a_1 = 3, ratio r = 1/3. Since |r| < 1, S = 3/(1 - 1/3) = 3/(2/3) = 4.5.
An infinite geometric series with common ratio r = -2 is convergent.
Show answer
Answer: False
For convergence, |r| must be less than 1. Since |−2| = 2 > 1, the series diverges.
The 10th term of the arithmetic sequence 5, 8, 11, 14, ... is ___.
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Answer: 32
a_n = 5 + (n-1)*3; a_10 = 5 + 9*3 = 5 + 27 = 32.
Which expression represents the sum of the first n terms of a geometric series with first term a_1 and ratio r (r ≠ 1)?
1. a_1 * n * r
2. a_1(1 - r^n)/(1 - r)
3. a_1 + (n-1)r
4. a_1/(1 - r)
Show answer
Answer: a_1(1 - r^n)/(1 - r)
The standard formula for the sum of a finite geometric series is S_n = a_1(1 - r^n)/(1 - r).
The sequence a_n = n/(n+1) converges to 1 as n approaches infinity.
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Answer: True
As n → ∞, n/(n+1) = 1/(1 + 1/n) → 1/(1+0) = 1, so the limit is 1.

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