Sequences | Free Calculus 2 Lesson (4 College Credits) | UPI Study

UPI Study › Courses › Calculus 2 › Free lessons › Lesson 6.1 Free lesson

Sequences

Calculus 2 · Chapter 6: Sequences & Introduction to SeriesSection 6.1 · lesson 27 of 5410 min · free

← The Logistic ModelNext lesson: Infinite Series and Partial Sums →

Earn 4 college credits for Calculus 2Recommended for credit by NCCRS · Math 265 · self-paced · one payment, lifetime access
Enroll in Calculus 2 · $250 →Download the syllabus (PDF)

In this lesson

Ordered lists of numbers, and where they are headed.

  • Describe a sequence explicitly or recursively
  • Decide convergence, borrowing L'Hopital's Rule when useful
  • Use boundedness and monotonicity to certify a limit exists
  • Apply the limit laws and the standard limit facts

Transcript

Everything said in the video, section by section (1,484 words). Read along, or search it.

MAT-265 - Calculus 2 · Section 6.1

Sequences

Deposit a fixed sum in an account and record the balance at the end of each year. The result is an ordered list of numbers. One per year. Measure the concentration of a drug in the bloodstream every hour after a dose. Or track the number of rabbits in a field each spring. The same structure appears every time. A list whose entries are indexed by the counting numbers. Such an ordered list is a sequence. And the central question about any sequence is this. Where is it headed? Does it settle toward a single value? Grow without bound? Or wander forever? Answering that question is the gateway to infinite series, Taylor polynomials, and much of the analysis used throughout science and engineering.

The Object

Explicit and Recursive Forms

Let's be precise. A sequence is an ordered list of numbers, with one term for each positive integer. Formally, a sequence is a function whose domain is the positive integers. The output a sub n is the nth term. And the whole sequence is written with braces around a sub n. One thing worth stressing. Order matters. A sequence is not merely a set — because each term occupies a definite position, indexed by n. There are two standard ways to describe a sequence. An explicit formula gives the nth term directly as a function of n. So any term can be computed without knowing the others. For example, a sub n equals one over n produces one, one half, one third, one quarter, and so on. A recursive formula instead defines each term using one or more previous terms, together with a stated starting value. For example, a one equals one, and a sub n equals two times a sub n minus one, produces one, two, four, eight, sixteen. The most famous recursive sequence is Fibonacci. F one and F two both equal one, and each later term is the sum of the two before it. One, one, two, three, five, eight, thirteen.

The Limit

Convergence and Divergence

The key question for a sequence is its long-run behavior. A sequence converges to a number L if its terms get arbitrarily close to L as n grows large — and stay close. The number L is called the limit of the sequence. If no such finite L exists — because the terms grow without bound, or oscillate, or otherwise fail to settle — the sequence diverges. Look at the two pictures, because they show exactly what that means. On the left, a sub n equals one plus one over n. The terms start at two, then drop toward one. And notice the shaded band. Eventually every term lies inside that thin band around L equals one. And they stay there. That's convergence. Pick any band, however narrow, and from some point onward every term is inside it. On the right, b sub n equals minus one to the n. The terms are minus one, one, minus one, one, forever. They're bounded — they never escape to infinity. But they never settle on a single value either. So the sequence diverges. Divergence doesn't only mean blowing up. It also means failing to settle.

The Bridge

Borrow the Tools You Already Have

Here's a bridge that saves an enormous amount of work. If a sub n equals f of n, for some function f, and the limit of f of x as x goes to infinity is L, then the limit of the sequence is also L. In other words: replace the discrete index n with a continuous variable x, and every technique from earlier calculus becomes available. Including L'Hopital's Rule. Let's use it. Determine whether a sub n equals log n over n converges, and if so, to what. Replace n with x and evaluate the limit of log x over x. Both numerator and denominator tend to infinity. That's the indeterminate form infinity over infinity. So L'Hopital applies. Differentiate top and bottom separately. You get one over x, over one. Which is one over x. And that goes to zero. Since the function limit is zero, the sequence converges — and its limit is zero. Notice we used a Chapter Two technique on a Chapter Six object. That bridge is worth remembering.

Structure

Bounded and Monotonic Sequences

Two structural properties help decide convergence even when the limit itself is hard to compute. A sequence is bounded above if there's a number M with every term at most M. Bounded below if there's a number little m with every term at least m. And bounded if both hold. A sequence is monotonic if it never changes direction. Either always increasing, or always decreasing. Now, those two ideas combine into a genuinely powerful guarantee. The Monotonic Sequence Theorem. Every bounded, monotonic sequence converges. Intuitively, think about why. A sequence that keeps rising but can never pass a ceiling has nowhere to go but toward a limit. It can't run off to infinity — the ceiling stops it. And being monotonic, it can't oscillate. The same holds for a falling sequence held above by a floor. And here's the theorem's real value. It certifies convergence without producing the limit. Which is essential for recursively defined sequences — where an explicit formula may be unavailable, but boundedness and monotonicity can still be established directly. Take a sub n equals n over n plus one. That's one half, two thirds, three quarters, four fifths. Bounded: every term is positive and less than one, since the numerator is smaller than the denominator. Monotonic: rewrite it as one minus one over n plus one. As n increases, the subtracted term shrinks, so the sequence increases. Increasing and bounded above by one. So it converges, by the theorem. And its limit is in fact one.

The Toolkit

Limit Laws and Standard Facts

Sequence limits obey the same algebraic laws as function limits. If two sequences converge, then their sum, difference, product and quotient converge to the sum, difference, product and quotient of the limits. With a nonzero denominator limit for division. And a constant multiple simply scales the limit. These laws let complicated sequences be broken into manageable pieces. Several standard limits recur constantly, and are worth memorizing outright. One over n to the p goes to zero, for any positive p. r to the n goes to zero whenever the absolute value of r is less than one. n to the one over n goes to one. And one plus x over n, all to the n, goes to e to the x. Also worth knowing: if the absolute value of r exceeds one, then r to the n diverges. And polynomial growth always loses to exponential growth. n to the k, over r to the n, goes to zero for r greater than one. Let's apply the laws. Find the limit of three n squared plus two n, over five n squared minus one. Divide numerator and denominator by the highest power. n squared. You get three plus two over n, over five minus one over n squared. Now apply the limit laws term by term, using one over n goes to zero and one over n squared goes to zero. You get three plus zero, over five minus zero. Three fifths. Dividing by the dominant power is the standard first move for any rational expression in n.

Key Takeaways

What to Carry Forward

Here's something to think about. The Monotonic Sequence Theorem guarantees that a bounded, monotonic sequence converges. But it does not say what the limit is. Consider the sequence defined recursively by a one equals two, and a sub n plus one equals one half, times the quantity a sub n plus two over a sub n. It can be shown to be decreasing, and bounded below by root two. So what does the theorem let you conclude? And how might you find the actual limit, once convergence is known? Here's a hint. If a sub n approaches L, then a sub n plus one also approaches L. So what equation must L satisfy? Five takeaways. A sequence is an ordered list indexed by the positive integers, described explicitly — a formula in n — or recursively, each term from earlier ones. A sequence converges to L if its terms settle arbitrarily close to L. Otherwise it diverges. And if a sub n equals f of n, and f has a limit at infinity, the sequence shares it. So L'Hopital's Rule applies. A sequence is bounded if its terms stay between fixed values, and monotonic if it never reverses direction. The Monotonic Sequence Theorem: every bounded, monotonic sequence converges — certifying a limit even without computing it. And sequence limits obey the usual limit laws. Memorize the standard facts: one over n to the p goes to zero, r to the n goes to zero for absolute r less than one, and n to the one over n goes to one. Next up is section six point two — Infinite Series and Partial Sums. Where adding a sequence up becomes a sequence of its own.

About Calculus 2 at UPI Study

Calculus 2 (Math 265) is a self-paced online course of 54 video lessons in 10 chapters, from integration techniques to series, parametric curves and vectors. It carries a NCCRS credit recommendation for 4 semester credits.

Every lesson video and transcript is free on pages like this one. Enrolling adds a practice quiz after each lesson, the graded work and final exam that earn the credit, lifetime access, and an official transcript on completion. UPI Study has helped over 130,000 students with college credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Enroll now → Course details Download the syllabus Every course with PRO →

Why take it here

Why students take Calculus 2 with UPI Study

Open the door to higher mathCalculus 2 is the gateway to Calculus 3, differential equations, statistics and most engineering, physics, economics and computer-science degrees. Completing it now lets you qualify for those courses at your university.
Credit at a fraction of the priceThe same 4 credits can cost $750 or more at a US college. Here it is one payment, self-paced, with lifetime access.
Finish your degree soonerTransfer credits reduce the courses you still owe, so you can graduate earlier or lighten a heavy semester.
Strengthen an applicationA completed college-level course on an official transcript shows admissions and scholarship committees you can do the work.
Study on your own scheduleNo start dates, no deadlines. Watch the lessons, practice, and take the final when you are ready.
130,000+students helped with college creditsACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Where the credit goes

Universities that work with NCCRS

Calculus 2 carries a NCCRS credit recommendation. These universities work with NCCRS; some are UPI Study's signed partners. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Thomas Edison State UniversityThomas Edison State UniversitySigned partnerUniversity of the PeopleUniversity of the PeopleSigned partnerCharter Oak State CollegeCharter Oak State CollegeSigned partnerStrayer UniversityStrayer UniversitySigned partnerColumbia Southern UniversityColumbia Southern UniversitySigned partnerIndiana TechIndiana TechSigned partnerPenn StatePenn StateWorks with NCCRSSouthern New Hampshire UniversitySouthern New Hampshire UniversityWorks with NCCRSSUNYSUNYWorks with NCCRS +many more cooperating universitiesFind my college →

Find my college → Not listed? Ask Athena in the corner of this page, or book a 15-minute call.

FAQ

Questions students ask us

About Sequences

What is the difference between an explicit and a recursive formula for a sequence?
An explicit formula gives the nth term directly as a function of n, while a recursive formula defines each term from one or more previous terms plus a starting value. For example, a_n = 1/n is explicit and gives 1, 1/2, 1/3, \dots. The recursion a_1 = 1, a_n = 2a_{n-1} gives 1, 2, 4, 8, \dots. Fibonacci, with F_1 = F_2 = 1 and F_n = F_{n-1} + F_{n-2}, is the classic recursive example.
How do you use L'Hopital's Rule to find the limit of a sequence?
Replace n with a continuous variable x, apply L'Hopital's Rule to \lim_{x\to\infty} f(x), and if that limit is L, the sequence a_n = f(n) also converges to L. For a_n = \ln n / n, the limit of \ln x / x has the indeterminate form \infty/\infty. Differentiating the top and bottom separately gives (1/x)/1 = 1/x, which tends to 0. So the sequence converges to 0.
What does the Monotonic Sequence Theorem say?
The Monotonic Sequence Theorem says every bounded, monotonic sequence converges. It certifies that a limit exists without computing it, which is especially useful for recursive sequences that lack an explicit formula. For example, a_n = n/(n+1) = 1 - 1/(n+1) is increasing because the subtracted term shrinks as n grows. It is also bounded above by 1. So the theorem guarantees convergence, and the limit turns out to be 1.
Does a bounded sequence have to converge?
No, a bounded sequence can still diverge if it fails to settle on a single value. The sequence b_n = (-1)^n alternates -1, 1, -1, 1, \dots and never leaves the interval [-1, 1], yet it has no limit. Divergence means more than blowing up to infinity, because oscillating without settling also counts. Boundedness guarantees convergence when it is combined with monotonicity, as stated in the Monotonic Sequence Theorem.

About the Calculus 2 course

How many college credits is Calculus 2?
Calculus 2 (Math 265) carries a NCCRS credit recommendation for 4 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
How much does Calculus 2 cost and what is included?
$250, paid once, with lifetime access: all 54 video lessons, a practice quiz after each, the graded work and final exam, and an official transcript when you finish. Students who want several courses choose UPI Study PRO instead.
Which universities accept UPI Study credits?
UPI Study has signed partner agreements with universities such as Thomas Edison State University, University of the People and Charter Oak State College, and works with universities such as Penn State, SNHU and SUNY. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities. Use "Find my college" on this page to check yours.
How long does Calculus 2 take?
It is self-paced with no deadlines. The 54 lessons total about 9 hours of video; most students add reading and practice time and finish in a few weeks to a few months, on their own schedule.
How are the exams done?
Quizzes and assignments are online in the course. The final exam is taken online with proctoring; a person on the UPI Study team checks assignments, usually within two days.
How do my credits reach my college?
When the course shows 100% complete, request the official transcript from your dashboard. It is sent to the college you name, and you can send it to more than one. Check your college first with "Find my college" on this page, or ask Athena.

Ready for the full course? Earn 4 credits in Calculus 2.

54 video lessons, a quiz after each one, and the graded work that earns the credit. Recommended for credit by NCCRS.

Enroll in Calculus 2 →Download the syllabusEvery course with PRO →
Calculus 2
4 credits · NCCRSEnroll →