Unit 17: Tests for Convergence
The full toolkit for deciding whether a series converges, derived rather than just asserted.
9 topics · 40 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Integral Test
- How it worksHow do rectangles drawn against a decreasing curve trap a series' sum between two related improper integrals?
- Worked exampleShow me whether the sum of 1/(n ln(n)) starting at n = 2 converges using the integral test.
- Worked exampleFind whether the sum of n e−n^2 converges by integrating the matching function with the integral test.
- Common mistakeWhy isn't the value of the matching improper integral equal to the value of the series' sum?
- Going furtherWhat conditions on the function have to hold before the integral test is even allowed?
Your turn1 of 2
Using the integral test, find the value of the integral from 2 to infinity of 4x e−x^2 dx (the improper integral that determines convergence for the sum of 4n e−n^2).The Direct Comparison Test
- How it worksHow do you pick a known series to compare against and set up the inequality that decides an unknown series' fate?
- Worked exampleShow me that the sum of 1/(n2 + 3) converges by comparing it to the sum of 1/n2.
- Worked exampleShow me that the sum of 1/(√n - 1) starting at n = 2 diverges by comparing it to the harmonic series.
- Common mistakeWhy doesn't a series being smaller than a divergent series tell you anything about its own convergence?
Your turn1 of 2
To show the sum of 1/(n2 + 3) converges, which comparison correctly proves it?The Limit Comparison Test
- How it worksHow does a positive, finite limit of an over bn rescue a comparison that the raw inequality can't cleanly make?
- Going furtherWhat happens when the limit in a limit comparison test comes out to infinity instead of a positive finite number?
- Worked exampleShow me that the sum of (2n + 1)/(n3 + 4) converges using limit comparison with the p-series 1/n2.
- Common mistakeWhy isn't a limit comparison result of zero automatically a useless outcome?
Your turn1 of 2
To show the sum of (2n + 1)/(n3 + 4) converges using limit comparison, which comparison series and limit are correct?The Alternating Series Test
- How it worksHow do shrinking terms with alternating signs squeeze the partial sums back and forth toward the limit?
- Worked exampleShow me that the alternating harmonic series 1 - 1/2 + 1/3 - 1/4 converges using the alternating series test.
- Worked exampleFind whether the series with terms (-1)n times n/(n2 + 3) passes the alternating series test.
- Common mistakeWhy doesn't the alternating series test apply the moment a series' terms stop decreasing?
Your turn1 of 2
Does the alternating series 1 - 1/2 + 1/3 - 1/4 + ... converge by the alternating series test?The Alternating Series Error Bound
- How it worksHow does the zigzag of consecutive partial sums trap the true value between them and bound the error?
- Worked exampleFind how many terms of 1 - 1/3 + 1/5 - 1/7 are needed to estimate the sum with error under 0.01.
- Worked exampleFind the error bound when the alternating harmonic series is cut off after 8 terms.
- Common mistakeWhy doesn't the alternating series error bound apply to a series where every term has the same sign?
Your turn1 of 2
Find the error bound when the series 1 - 1/3 + 1/5 - 1/7 + ... is cut off after 8 terms.Absolute Versus Conditional Convergence
- How it worksHow does testing the series of absolute values first sort every convergent series into an absolute or conditional bucket?
- Worked exampleShow me that the sum of (-1)n divided by n converges conditionally but not absolutely.
- Worked exampleShow me that the sum of (-1)n divided by n2 converges absolutely.
- Common mistakeWhy doesn't rearranging the order of a conditionally convergent series's terms preserve its original sum?
- Going furtherWhat happens to a series' terms when it converges absolutely instead of just conditionally?
Your turn1 of 2
Does the sum of (-1)n divided by n converge conditionally, absolutely, or not at all?The Ratio Test
- How it worksHow does the limit of the ratio of consecutive terms decide convergence, divergence, or nothing at all?
- Worked exampleShow me what the ratio test says about the convergence of the sum of 2n divided by n factorial.
- Worked exampleFind what the ratio test concludes about the sum of n2 divided by 3n.
- Common mistakeWhy doesn't the ratio test decide anything at all when its limit comes out to exactly 1?
Your turn1 of 2
What does the ratio test say about the convergence of the sum of 2n divided by n factorial?The Root Test
- How it worksHow does the limit of the nth root of the nth term decide convergence the same way the ratio test's limit does?
- Worked exampleShow me that the sum of (n/(2n + 1))n converges using the root test.
- Worked exampleFind what the root test concludes about the sum of 3n divided by nn.
- Common mistakeWhy doesn't the root test need to worry about factorials the way the ratio test's cancellation trick does?
- Ties togetherHow does the root test relate to the ratio test when both are applied to a series with an nth-power structure?
Your turn1 of 2
Does the sum of (n/(2n + 1))n converge, using the root test?Choosing a Convergence Test
- How it worksHow do you read the shape of a series' general term to decide between comparison, the integral test, and the ratio test?
- Worked exampleShow me which test cracks the sum of (n + 1)/(n2 + 3n) and why the others stall on it.
- Worked exampleFind the right test for the sum of (-1)n times n divided by 2n and then run it.
- Common mistakeWhy isn't the ratio test a safe default to try first on every series you encounter?
- Going furtherWhat happens when a series needs two different tests chained together, like limit comparison then alternating?
Your turn1 of 2
Which test cracks the sum of (n + 1)/(n2 + 3n), and why do the others stall on it?