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Strategy for Testing Series

Calculus 2 · Chapter 7: Convergence Tests for SeriesSection 7.7 · lesson 38 of 549 min · free

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In this lesson

Choosing the right test on the first attempt.

  • Diagnose a term's structure before computing anything
  • Work the decision sequence in cost order
  • Use the growth hierarchy to spot the dominant piece
  • Know that an inconclusive result is still information

Transcript

Everything said in the video, section by section (1,308 words). Read along, or search it.

MAT-265 - Calculus 2 · Section 7.7

Strategy for Testing Series

You now have seven convergence tests. The nth-Term Test. The p-series and geometric rules. The Alternating Series Test. The comparison tests. The Ratio and Root Tests. And the Integral Test. And the remaining difficulty is not executing any one of them. It's choosing which one to attempt. A wrong choice is rarely fatal. But it costs time, and it often produces an inconclusive result that then has to be abandoned. So this section is about developing the diagnostic habit. Read the structure of the general term first. Then pick the test.

The Skill

Diagnose Before You Compute

The single most useful thing to know when diagnosing a term is the growth hierarchy. Which building blocks grow faster than which. Look at the picture. Four common growth classes, plotted on a logarithmic vertical axis — so each curve's slope reflects its growth rate. The logarithmic curve, log n, is nearly flat. It barely grows at all. The polynomial curve, n squared, rises steadily. The exponential, two to the n, rises faster still. And the factorial outpaces everything. So the ordering is: logarithmic, then polynomial, then exponential, then factorial. Each one dominating the last. Notice where the exponential and factorial curves cross. Around n equals five. Beyond that point, factorial growth dominates decisively. Which is exactly why terms containing n factorial almost always call for the Ratio Test. Knowing this hierarchy lets you identify the dominant piece of a term at a glance.

The Order

Cheap Tests First

The tests are not interchangeable. And the order in which you attempt them matters. The sequence is arranged by cost. Cheap tests that frequently settle the question come first. Expensive tests that demand ingenuity come last. Step one. The nth-Term Test. If the limit of a n is not zero, the series diverges. This requires only a limit, and can be checked mentally in most cases. But remember — it never proves convergence. Step two. Recognition. If the term is one over n to the p, apply the p-series rule. If it's a times r to the n minus one, apply the geometric rule. No computation beyond identification is needed. This is the cheapest possible win. Step three. Sign alternation. If the term is minus one to the n, times a positive b n, apply the Alternating Series Test. Then examine the absolute-value series to classify the convergence as absolute or conditional. Step four. Rational or algebraic form. If the term is built from powers and roots of n, use the Limit Comparison Test — against the p-series matching the dominant powers. Step five. Factorials or nth powers. If n factorial or a base to the n appears, use the Ratio Test. And if the entire term is an nth power, the Root Test is usually simpler. Step six. Integrable form. If the term is f of n, with f positive, continuous and decreasing, and the integral of f is tractable, use the Integral Test. Step seven. Direct Comparison. Reserved for terms that resist all of the above. Typically those containing bounded oscillating factors, like sine n or cosine n.

Figure 7.7.2

The Decision Flowchart

The flowchart puts that sequence into a picture. Work top to bottom. Each diamond asks a structural question about the general term. And each box on the right names the resulting test. Note that the first question — does the term go to zero — can only produce divergence. Never convergence. And Direct Comparison sits last, deliberately. Because it requires you to supply a comparison series yourself, rather than generating one mechanically. That's the most expensive kind of work. One more thing about reading the chart. A no answer never closes off an earlier branch. Algebraic simplification of the term may reveal a structure that an earlier question would have recognized.

Reading the Chart

Three Things Worth Emphasising

Three features of this chart deserve emphasis. First. It's a heuristic, not an algorithm. More than one test often succeeds. The chart merely selects a route that usually succeeds quickly. Second. An inconclusive result is genuine information. It eliminates a branch and directs your attention elsewhere. It is not a failure of method. Third. Algebraic preparation frequently changes the route. Take the term two to the n plus three to the n, all over four to the n. At first glance that appears to require the Ratio Test. But split the fraction. You get two to the n over four to the n, plus three to the n over four to the n. Which is one half to the n, plus three quarters to the n. Two geometric series. Both with ratio less than one. Settled by recognition alone. No test computation at all. Let's do a full diagnosis. Determine whether the sum of n squared plus five n, over four n cubed minus n plus seven, converges. Step one. Check the nth-Term Test. Numerator degree two, denominator degree three. So a n goes to zero. The nth-Term Test gives no information. That's the expected outcome here, not a failure. Step two. Identify the structure. It's a ratio of polynomials. The flowchart routes rational expressions to Limit Comparison. The dominant behavior is n squared over four n cubed. Which is one over four n. So compare with b n equal to one over n. Step three. Compute the limit. You get n cubed plus five n squared, over four n cubed minus n plus seven. Divide by n cubed. The limit is one quarter. Finite and strictly positive. So the Limit Comparison Test applies, and the two series share a common fate. Since the sum of one over n is the harmonic series, which diverges — the given series diverges.

Reference

Term Forms and the Test of First Resort

The table condenses the flowchart into a lookup format. Each recognizable term structure, paired with the test of first resort, and a worked instance. Terms not going to zero: nth-Term Test. Diverges. One over n to the p: p-series rule. A times r to the n minus one: geometric rule. Alternating with magnitudes decreasing to zero: Alternating Series Test. Ratio of polynomials: Limit Comparison. Roots and powers of n: also Limit Comparison. Contains a factorial: Ratio Test. Entire term is an nth power: Root Test. Positive decreasing f of n, with a tractable integral: Integral Test. And a bounded oscillating factor: Direct Comparison. One caution about the table. It's a starting point, not a guarantee. The entry for one over root n squared plus one, for instance, compares with the harmonic series and diverges. Even though the presence of a square root might suggest otherwise at a glance.

Key Takeaways

What to Carry Forward

One more example, to show how fast the right diagnosis makes things. Determine whether the sum of five to the n over n factorial converges. Look at the structure. A factorial in the denominator, and a fixed base raised to the nth power in the numerator. That sends it directly to step five of the sequence. The Ratio Test. Form the ratio. All terms are positive, so absolute values may be omitted. Simplify, using five to the n plus one equals five to the n times five. And n plus one factorial equals n plus one, times n factorial. Almost everything cancels. You're left with five over n plus one. Which goes to zero. Less than one. So the series converges absolutely. One test. Three lines. Five takeaways. Read the structure before computing. The growth hierarchy identifies the dominant piece. Work the sequence in cost order. nth-Term Test, recognition, alternation, rational form, factorials and powers, integrability, then direct comparison. The nth-Term Test never proves convergence. And recognition of a p-series or geometric series needs no computation at all. An inconclusive result eliminates a branch. It's information, not failure. And algebraic simplification can change the route entirely — turning an apparently hard series into one settled by recognition. That closes Chapter Seven. Next up is Chapter Eight — Power Series and Taylor Series, where these very tests decide where a function's series representation is valid.

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FAQ

Questions students ask us

About Strategy for Testing Series

How do you decide which convergence test to use on a series?
Read the structure of the general term first, then try the tests in order of cost. Start with the nth-Term Test, then look for a p-series or geometric series, then alternating signs. Use Limit Comparison for rational or algebraic terms and the Ratio or Root Test for factorials and nth powers. Try the Integral Test for a tractable positive decreasing f(n), and save Direct Comparison for last, typically for factors like \sin n.
What is the growth hierarchy and how does it help test series?
The growth hierarchy ranks common building blocks from slowest to fastest growing: logarithmic, polynomial, exponential, then factorial, so eventually \ln n \ll n^2 \ll 2^n \ll n!. It lets you spot a term's dominant piece at a glance. For example, in \sum 5^n/n! the factorial dominates, which points to the Ratio Test. The ratio simplifies to 5/(n+1) \to 0 < 1, so the series converges absolutely.
Does the series \sum \frac{n^2+5n}{4n^3-n+7} converge or diverge?
It diverges, by Limit Comparison with the harmonic series. The nth-Term Test gives no information because the terms go to 0. The dominant behavior is n^2/(4n^3) = 1/(4n), so compare with b_n = 1/n. Then \lim a_n/b_n = \lim \frac{n^3+5n^2}{4n^3-n+7} = \frac14, which is finite and positive. Both series share the same fate, and \sum 1/n diverges.
If the terms of a series go to zero, does the series converge?
No, a limit of 0 makes the nth-Term Test inconclusive, never a proof of convergence. The harmonic series \sum 1/n has terms going to 0 but diverges. An inconclusive result is still useful information because it eliminates a branch and sends you to another test. Also, try simplifying algebraically. For example, \frac{2^n+3^n}{4^n} = (\frac12)^n + (\frac34)^n splits into two convergent geometric series.

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