Calculus II ep21: Boundedness, series (Oct 29, 2026)

Overview

The lecture reviews monotonic and bounded sequences, establishes that every convergent sequence is bounded, and introduces the Monotone Convergence Theorem: a bounded monotonic sequence converges. It then defines an infinite series as the limit of its partial sums, illustrating convergence with the geometric series whose terms are 1/2^n and divergence with both the constant-one series and the alternating series with terms (-1)^n.

Key takeaways

A bounded sequence need not converge: (-1)^n remains within [-1, 1] but oscillates indefinitely.
Every convergent sequence is bounded because its tail stays near its limit and its finite initial segment has a finite maximum and minimum.
The Monotone Convergence Theorem guarantees convergence when a sequence is both monotonic and bounded, even if it never reaches its limiting value.
An infinite series is defined through partial sums: the series converges exactly when the sequence of finite sums approaches a finite limit.
The geometric series 1/2 + 1/4 + 1/8 + ... converges to 1, while the constant-one series diverges because its partial sums equal n.
The alternating series -1 + 1 - 1 + 1 + ... diverges because its partial sums alternate between -1 and 0 instead of approaching one value.

Chapters

0:00 Monotonic Sequences and the Example (-n)^n

A sequence is monotonic if it is always increasing or always decreasing; being not increasing does not automatically mean it is decreasing.
The terms of (-n)^n begin -1, 4, -27, and 256, so the sequence is neither increasing nor decreasing.
Writing out initial terms can identify monotonicity when a formal proof is not required.

3:00 Upper Bounds, Lower Bounds, and Bounded Sequences

A sequence is bounded above if some real number is at least as large as every term; that number is an upper bound.
A lower bound is a number no greater than any term, and a sequence is bounded when it has both an upper and a lower bound.
Boundedness means all sequence values fit within a fixed horizontal interval.

7:00 Testing Bounds with n/(n²+1), 3^n, and (-1)^n

For n/(n²+1), every term is positive, so 0 is a lower bound; n²+1 exceeds n for positive integers, so 1 is an upper bound.
The sequence 3^n is bounded below, for example by 0, but is not bounded above because it tends to infinity.
The sequence (-1)^n alternates between -1 and 1, so bounds such as -2 and 2 show it is bounded.

15:00 Why Boundedness and Convergence Are Different

Boundedness alone does not imply convergence: (-1)^n stays between -1 and 1 but keeps alternating.
Every convergent sequence is bounded: its terms eventually stay close to the limit, while the finite set of earlier terms also has a bound.
A sequence cannot have infinitely large early values before settling down, because it has only finitely many terms before any fixed index.

23:00 The Monotone Convergence Theorem

The theorem states that a sequence that is both monotonic and bounded must converge.
A bounded increasing sequence approaches a limiting value no greater than its upper bound; it need not reach the upper bound.
The same result applies to bounded decreasing sequences, which approach a limiting value within their bounds.

27:00 Series as Infinite Sums, Including a Geometric Example

A series is the sum of all terms in a sequence, commonly written with sigma notation or as an unbounded sum.
An infinite series does not necessarily equal infinity: the geometric series 1/2 + 1/4 + 1/8 + ... sums to 1.
The geometric series with terms 1/3^n is raised as a related example whose sum requires a method beyond the simple area-filling illustration.

33:00 Defining Series with Sequences of Partial Sums

For terms a_n, the partial sums are s₁ = a₁, s₂ = a₁ + a₂, and generally s_k = a₁ + ... + a_k.
The infinite series is defined as lim n→∞ s_n, turning the question of summing infinitely many terms into a sequence-limit question.
A series converges when its partial sums have a finite limit and diverges when that limit does not exist.

36:00 The Constant-One Series Diverges

For a_n = 1, the underlying sequence converges to 1 because every term is 1.
The partial sums are s₁ = 1, s₂ = 2, s₃ = 3, and generally s_n = n.
Since lim n→∞ n is infinite rather than finite, the series 1 + 1 + 1 + ... diverges.

40:00 Why the Alternating Series (-1)^n Does Not Sum to Zero

The series with terms (-1)^n begins -1 + 1 - 1 + 1 + ..., which can misleadingly suggest that terms cancel to zero.
Its partial sums alternate between -1 for odd n and 0 for even n, so the partial-sum sequence has no limit.
Because its partial sums do not converge, the alternating series diverges; it does not sum to either 0 or -1.

Transcript excerpt

0:00 uh where the business really gets going. So uh let's talk a little bit about se uh sequences uh just one last time. Remember I will refresh our memory the definition. Uh remember monotonic means either increasing or decreasing. Either increasing or decreasing. That's called monotonic. In ordinary language, people might say the opposite of increasing is decreasing. But that's not really true. Like logically speaking, the opposite of increasing is not increasing. And that doesn't mean that it's decreasing. You know, not increasing is not the same as decreasing. Um, if it's increasing or decreasing, we say it's monotonic. And

0:48 if it's not, we say it's not monotonic. That means that it's not always increasing and also not always decreasing. Uh so for example uh a n equ= minus n ^ n is that increasing or decreasing or neither? Um if I ask you to prove it that's a bit of work. If I just ask you is it increasing or decreasing? It's good enough to just write out a few terms and see see for yourself. Is it increasing or or decreasing? If I'm not asking you to show it. Um, is it uh ink or deck or neither? It could be neither. Uh, let's just write out a few terms. We um, unless I say so, you should start with n equals 1. So, this would be first term

1:36 minus1 to the 1 and then minus 2 to the 2, -3 to the 3, etc. It's probably good enough to get an idea. And now uh let's just say sort of what those are. Minus one to the one is one uh minus one sorry you have to pay attention to the exponents. Uh -2 to the two because this is negative but being squared we're going to have we're going

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