Sum of Geometric Series | Free Discrete Mathematics Lesson (4 College Credits) | UPI Study

UPI Study › Courses › Discrete Mathematics › Free lessons › Lesson 3.7 Free with a free account

Sum of Geometric Series

Discrete Mathematics · Chapter 3: Sequences and SeriesSection 3.7 · lesson 23 of 879 min · free with a free account

← Sum of Arithmetic SeriesNext lesson: Practice: Arithmetic and Geometric Series →

Earn 4 college credits for Discrete MathematicsRecommended for credit by NCCRS · Math 230 · self-paced · one payment, lifetime access
Enroll in Discrete Mathematics · $250 →Download the syllabus (PDF)

In this lesson

Adding finitely many terms - and when infinitely many still add to a finite number.

  • Derive the finite geometric series formula
  • Compute partial sums in closed form
  • Define convergence of an infinite series
  • Apply the sum-to-infinity formula, and its condition

Transcript

Everything said in the video, section by section (1,342 words). Read along, or search it.

Sequences and Series

Sum of Geometric Series

Section three point four studied geometric sequences term by term. This section adds them up. We'll derive a formula for summing finitely many terms of a geometric sequence. Then we'll find a precise condition under which summing infinitely many of them still produces a finite number.

The Opening Problem

How Long Has the Client Waited?

Section three point four looked at the exponential-backoff retry policy, with a first delay of five hundred milliseconds and a common ratio of two. It asked how long the wait is before the tenth retry alone. That question has a one-term answer. A different, equally practical question doesn't. How much total time has the client spent waiting, once all its retries are added up? That asks for a sum of terms, a partial sum of a geometric sequence. And section three point four also had a shrinking timeout budget, starting at eight thousand milliseconds, with a ratio of one half. If that pattern were extended forever, would its total be finite? Or would it also grow without bound?

Partial Sums

From Terms to Sums

For a sequence, the n-th partial sum, S sub n, is the sum of its first n terms. A series is that running total, studied as its own object. Section three point six summed arithmetic series by pairing the sequence with its own reverse. That trick won't help here. It works only because consecutive arithmetic terms differ by a fixed amount. Consecutive geometric terms differ by a fixed factor instead. So summing them calls for a different technique.

Derivation

Multiply by r, Then Subtract

Let S sub n be the sum of the first n terms. a one, plus a one r, plus a one r squared, up to a one r to the n minus one. Multiply both sides by r. That gives a one r, plus a one r squared, up to a one r to the n. Now subtract the second equation from the first. Every term from a one r up to a one r to the n minus one appears in both sums, and cancels. Only the first term of the first sum, and the last term of the second, survive. So S sub n minus r S sub n equals a one minus a one r to the n. Factor both sides: one minus r, times S sub n, equals a one times one minus r to the n.

Theorem

The Finite Geometric Series Sum

Divide by one minus r, provided r isn't one. The theorem. The sum of the first n terms of a geometric sequence is a one, times one minus r to the n, all over one minus r. If r equals one, the denominator is zero, and the formula doesn't apply. But in that case every term equals a one, so the sum of n terms is simply n times a one.

Worked Example 1

Total Wait After 5 Retries

Let's find the total wait after five retries, with first delay five hundred and ratio two. S five equals five hundred, times one minus two to the fifth, over one minus two. That's five hundred times minus thirty-one, over minus one. Fifteen thousand five hundred milliseconds. Check it by direct addition. Five hundred, one thousand, two thousand, four thousand, and eight thousand add to fifteen thousand five hundred. It matches. The closed form also changes the cost. Adding n terms one at a time takes work that grows with n. The formula needs one exponentiation and a few operations, which, with an efficient algorithm, grows only with the logarithm of n.

Definition

When Does an Infinite Sum Make Sense?

Now suppose the number of terms isn't fixed at all. The sequence keeps going forever. It's not obvious that adding infinitely many positive numbers could ever produce a finite result. For many sequences, it doesn't. So the sum of an infinite series is given a precise meaning, as a limit. An infinite series converges to a sum S, if its partial sums approach S as n grows without bound. If the partial sums don't approach one finite value, because they grow without bound or oscillate without settling, the series diverges, and no sum is assigned.

Sum to Infinity

Everything Hinges on r to the n

Start from the finite sum formula, and watch what happens as n grows. Everything in the formula is fixed except r to the n. So the behavior of the sum depends entirely on the behavior of r to the n. If the absolute value of r is less than one, r to the n shrinks toward zero, even for negative r. Substituting zero gives a one over one minus r. If the absolute value of r is greater than one, r to the n grows without bound, so the series diverges. If r equals one, the sum is n times a one, which grows without bound. And if r equals minus one, the partial sums swing between a one and zero forever. Both diverge.

Theorem

The Sum to Infinity

Putting the cases together gives a single, precise condition. An infinite geometric series converges if and only if the absolute value of r is less than one. When it converges, its sum is a one over one minus r. When the absolute value of r is one or more, the series diverges, and no finite sum exists.

Worked Example 2

Does the Budget Add Up to Something Finite?

Take the timeout budget, starting at eight thousand milliseconds with ratio one half. Suppose, hypothetically, the pattern were extended indefinitely. Check convergence first. The absolute value of r is one half, which is less than one. So the series converges. Apply the formula. Eight thousand over one minus one half is eight thousand over one half, which is sixteen thousand milliseconds. Sixteen seconds. Even though this policy never stops proposing another, smaller allotment, the total is bounded. However many terms are included, the running total never reaches sixteen seconds. It only approaches it.

Figures 1 and 2

Divergent Versus Convergent

This figure compares the two series. On the left are the running totals of the retry delays, for one through six retries. Each bar is a cumulative wait, not a single delay. They climb ever more steeply, rather than leveling off. That's the signature of a series with no sum to infinity. On the right are the running totals of the timeout budget. They flatten toward the dashed line at sixteen thousand milliseconds, without ever reaching it. That's the qualitative difference between a divergent and a convergent series.

The Engineering Distinction

Growing Delays Have No Finite Total

For the retry delays, asking for the total if the retries never stop isn't a matter of computing a bigger number. It isn't a valid question. The absolute value of r is two, which is at least one, so the series diverges. So a retry policy whose wait times grow geometrically has no finite worst-case total wait, if it's allowed to continue indefinitely. One whose wait times decay geometrically does. That's a genuine engineering distinction, and it has the same mathematical root as the condition, the absolute value of r less than one.

Comparing 3 Kinds of Series

Only One Can Be Summed to Infinity

This table places three kinds of series side by side. The arithmetic series from section three point six needs the first term, the last term or the difference, and n. Its terms never shrink, so it grows without bound. The finite geometric series needs the first term, r, and n. It's finite for any finite n. And the infinite geometric series needs only the first term and r. n plays no role, because there's no last term. It converges to a finite value only if the absolute value of r is less than one. Otherwise no sum exists at all.

Takeaways

Telescoping, and a Condition

To summarize. The finite geometric series formula comes from multiplying by r and subtracting, so the middle terms cancel. That's a different technique from the arithmetic pairing trick. An infinite geometric series converges if and only if the absolute value of r is less than one. Then its sum is a one over one minus r. And terms shrinking toward zero is what makes convergence possible, but the sum itself generally isn't zero. Here's something to think about. A classmate says, the terms shrink toward zero, so adding infinitely many of them should give zero. But the budget example gave sixteen thousand milliseconds. Where does the reasoning go wrong? Be precise about the difference between a claim about the terms, and a claim about the partial sums.

About Discrete Mathematics at UPI Study

Discrete Mathematics (Math 230) is a self-paced online course of 87 video lessons in 13 chapters, from logic and proof to counting, probability, graph theory and trees. It carries a NCCRS credit recommendation for 4 semester credits.

Every lesson transcript is free on pages like this one, and the videos are free with a free UPI Study account. 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 Discrete Mathematics with UPI Study

The math behind computer scienceDiscrete Mathematics is the mathematics of computer science: logic, proofs, sets, counting, probability and graphs underpin algorithms, data structures, cryptography and databases. 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

Discrete Mathematics 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 Sum of Geometric Series

How do you derive the formula for the sum of a finite geometric series?
Multiply the sum by r and subtract the result from the original sum. Writing S_n = a_1 + a_1r + \dots + a_1r^{n-1}, the middle terms cancel, leaving (1-r)S_n = a_1(1-r^n). Dividing by 1-r gives S_n = a_1\frac{1-r^n}{1-r} for r \neq 1.
What is the sum of the first 5 terms of a geometric sequence with first term 500 and ratio 2?
The sum is 15,500. Using S_n = a_1\frac{1-r^n}{1-r} gives S_5 = 500\cdot\frac{1-2^5}{1-2} = 500\cdot\frac{-31}{-1} = 15{,}500. Direct addition confirms it: 500 + 1000 + 2000 + 4000 + 8000 = 15{,}500.
When does an infinite geometric series converge, and what is its sum?
An infinite geometric series converges exactly when |r| < 1, and then its sum is \frac{a_1}{1-r}. This works because r^n shrinks toward zero in the finite formula, even for negative r. If |r| \ge 1, the partial sums grow without bound or oscillate, so the series diverges and has no sum.
If the terms of a geometric series shrink toward zero, why isn't the infinite sum zero?
The terms approaching zero describes the terms, while the sum describes the partial sums, which keep accumulating. For a_1 = 8000 and r = \tfrac12, the sum is \frac{8000}{1-\frac12} = 16{,}000, not zero. Shrinking terms make convergence possible, but the sum is generally a nonzero number.

About the Discrete Mathematics course

How many college credits is Discrete Mathematics?
Discrete Mathematics (Math 230) 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 Discrete Mathematics cost and what is included?
$250, paid once, with lifetime access: all 87 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 Discrete Mathematics take?
It is self-paced with no deadlines. The 87 lessons total about 13 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 Discrete Mathematics.

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

Enroll in Discrete Mathematics →Download the syllabusEvery course with PRO →
Discrete Mathematics
4 credits · NCCRSEnroll →