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Properties of Convergent Series
Calculus 2 · Chapter 6: Sequences & Introduction to SeriesSection 6.5 · lesson 31 of 5410 min · free
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In this lesson
The algebra of infinite sums - and its limits.
- Factor constants out and add series term by term
- See why the converse of linearity fails
- Know that finitely many terms never affect convergence
- Avoid applying linearity to divergent series
Transcript
Everything said in the video, section by section (1,575 words). Read along, or search it.
MAT-265 - Calculus 2 · Section 6.5
Properties of Convergent Series
You now know how to sum two families of series exactly. Geometric, and telescoping. But most series you'll meet are neither. So before we start building convergence tests, it's worth establishing the algebra of infinite sums. What operations are you allowed to perform on a series? And which ones preserve convergence? The answers are mostly what you'd expect. But there are two places where intuition fails badly. And those two places are where students lose the most marks.
Linearity
Constants Factor Out, Series Add
The first property is linearity. If the sum of a n converges to A, and the sum of b n converges to B, and c is any constant, then two rules hold. First. The sum of c times a n equals c times the sum of a n. Constants factor out. Second. The sum of a n plus or minus b n equals A plus or minus B. Series add term by term. Both follow directly from the corresponding limit laws, applied to the partial sums. Which makes sense — a series' sum is by definition the limit of its partial sums, so anything true of limits transfers. Let's use them. Evaluate the sum of three times one half to the n, plus five times one third to the n, starting at n equals zero. By linearity, split the combined series into two separate geometric series and add their sums. The first has a equals three and r equals one half. So it sums to three over one half. Six. The second has a equals five and r equals one third. So it sums to five over two thirds. Fifteen halves. Add them. Six plus fifteen halves is twenty-seven halves. Second example. Evaluate the sum of seven over two to the n, starting at n equals one. Factor the constant seven out, leaving a geometric series. That remaining series starts at n equals one. So its first actual term is one half, and r is one half. Seven times one half over one half. Seven times one. Seven.
A Caution
Linearity Runs One Way Only
Now here's the first place intuition fails. Linearity runs in one direction only. If both series converge, their term-by-term sum converges. That direction is guaranteed. But the reverse is not. If the sum of a n plus b n converges, it does not follow that the individual series converge. Here's a clean counterexample. Let a n be one, and b n be minus one, for every n. Each of those series diverges — one grows without bound, the other falls without bound. Yet their term-by-term sum is the sum of zero. Which converges, to zero. So the linearity theorem requires the individual series to converge before the rules may be applied. There is one useful consequence worth stating on its own. If the sum of a n converges, and the sum of b n diverges, then the sum of a n plus b n diverges. The reason is a short proof by contradiction. If that combined sum converged, then subtracting the convergent sum of a n — using linearity — would force the sum of b n to converge. Contradicting the fact that it diverges. So a convergent series can never rescue a divergent one.
The Tail Is What Matters
Finitely Many Terms Never Affect Convergence
The second major property concerns what happens when a series is altered at the front. And the result is one of the most useful facts in the chapter. Changing, inserting, or deleting any finite number of terms cannot change whether a series converges. Though it generally does change the sum. Formally: for any positive integer M, the series and its tail starting at M plus one either both converge, or both diverge. Let's see why, because the reasoning is simple and clarifying. Split off the first M terms. That writes the full series as a finite sum, plus its tail. The bracketed piece — a one through a M — is just a fixed number. Call it P. Now, adding a constant P to the tail cannot change whether the tail's partial sums approach a limit. It merely shifts that limit by P. So the full series converges exactly when its tail does. And the two sums differ by P. And because M can be any finite number, no finite collection of terms ever determines convergence. Only the infinitely long tail does. This explains a convenience we've already been using informally. When testing a series for convergence, the starting index does not matter. And any finite number of troublesome early terms can simply be ignored. Convergence is a property of the tail behavior of a series. How its terms behave far out, as n goes to infinity.
Worked Examples 3 and 4
Same Convergence, Different Sum
Two examples to make the distinction sharp. The geometric series sum of one half to the n, from n equals zero, converges to two. What happens to convergence, and to the sum, if the first two terms are removed? Removing one and one half leaves the tail starting at n equals two. That's one quarter plus one eighth, and so on. This tail still converges — it's itself geometric, with a equal to one quarter and r equal to one half. Its sum is one quarter over one half. One half. So convergence is preserved. But the sum dropped from two to one half. Precisely because the removed terms, one plus one half, equal three halves. And two minus three halves is one half. The removal changed the value, but not the fact of convergence. Second example. The harmonic series diverges. Does prepending the terms one hundred plus fifty, or deleting the first thousand terms, change that? No. Adding two finite terms at the front, or deleting the first thousand, alters only a finite portion. The infinitely long tail behaves exactly as before, and still diverges. So the modified series diverges too. No finite change can convert a divergent series into a convergent one. Or the other way round. The table summarizes it. Multiply every term by c: convergence preserved, sum multiplied by c. Add two convergent series term by term: preserved, sums add. Remove finitely many terms: preserved, sum changes by the removed amount. Add finitely many terms: preserved, sum changes by the added amount. In every row, convergence is preserved. Only the value moves.
Together
Split, Sum, and Adjust the Index
The real power of these properties appears when they're combined. A complicated-looking series can often be broken into pieces, each recognized as a known convergent series, then reassembled by linearity. With any awkward leading terms split off separately. Let's do one that uses both properties. Evaluate the sum of one half to the n, plus one quarter to the n, starting at n equals zero. Then find the same sum starting at n equals one. By linearity, the first sum splits into two geometric series, each with first term one. One over one minus one half, plus one over one minus one quarter. That's two, plus four thirds. Ten thirds. For the second sum, which starts at n equals one, simply remove the n equals zero term. That term is one half to the zero, plus one quarter to the zero. One plus one. Two. By the finitely-many-terms property, the series still converges. And its sum is the full sum minus the removed term. Ten thirds minus two. Which is ten thirds minus six thirds. Four thirds. Both properties were used. Linearity to split and sum. And the finite-term rule to adjust the starting index. Now here's something to think about. The finitely-many-terms principle says convergence depends only on the tail, not on any finite front portion. Yet the principle draws a sharp line at the word finite. Why does the argument — the removed terms sum to a single fixed number P — break down completely if you try to remove infinitely many terms? What exactly goes wrong with the reasoning in that case?
Key Takeaways
The Pitfall, and What to Carry Forward
The most consequential error in this section is invoking linearity when one or both series diverge. The linearity theorem has an explicit hypothesis. Both series must converge. And the rules are simply false without it. A tempting but invalid move is to split a divergent series into two pieces, evaluate each as if convergent, and combine. For instance, you cannot write the sum of one minus one, as the sum of one, minus the sum of one — and conclude the answer is infinity minus infinity equals zero. The two pieces diverge, so the subtraction is meaningless. Even though the original sum of zero does converge to zero. The safeguard is to confirm convergence of each individual series first. And only then apply linearity. A related slip is assuming that because a combined series converges, its parts must. That's the converse we warned about earlier. When in doubt, establish convergence of the building blocks before manipulating them. Five takeaways. Convergent series are linear. Constants factor out, and convergent series add term by term. Linearity requires the individual series to converge first. The converse fails — a convergent combined series need not have convergent parts. A convergent plus a divergent series always diverges, since a convergent series cannot rescue a divergent one. Adding or removing finitely many terms never changes convergence, because the removed portion is a single fixed number. Only the infinite tail matters. And removing or adding terms does change the sum of a convergent series — by exactly the amount removed or added. That closes Chapter Six. Next up is Chapter Seven — Convergence Tests for Series, beginning with the Integral Test and p-series.
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