Calc 2 Series- Convergence Tests and Techniques

What Convergence Actually Means

A series is just a sum of infinitely many terms. Convergence is when those infinite terms add up to a finite number. Divergence is when they don't. That's it. No philosophical debates here.

Before you run any test, ask yourself: does the nth term go to zero? If it doesn't, the series diverges. That's the first and easiest check. The nth term test is a one-way door — passing it tells you nothing, but failing it ends the discussion.

The Tests (And When to Use Each)

There are seven tests you need to know. Each works in specific situations. Using the wrong test wastes time and leads to wrong answers.

Nth Term Test (Test for Divergence)

If lim(n→∞) aₙ ≠ 0, the series diverges.

This test only proves divergence. It cannot prove convergence. If the terms don't shrink to zero, you're done — the series diverges. If they do shrink to zero, keep testing.

Integral Test

Compare a series to an improper integral. If f(x) is continuous, positive, and decreasing, then:

• ∫ f(x) dx converges → series converges
• ∫ f(x) dx diverges → series diverges

Use this when you have a function you can integrate. P-series, harmonic series, and anything with factorials or exponentials in the denominator work well here.

Direct Comparison Test

Find a series you know converges (or diverges) and compare term-by-term.

If aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges.

If aₙ ≥ bₙ and Σbₙ diverges, then Σaₙ diverges.

This test requires intuition. You need to find a comparable series. The comparison is only as good as your choice of the test series.

Limit Comparison Test

Take the limit of the ratio between your series and a known series:

lim(n→∞) aₙ/bₙ = L

• If L is finite and nonzero → both series behave the same
• If L = 0 and Σbₙ converges → Σaₙ converges
• If L = ∞ and Σbₙ diverges → Σaₙ diverges

This test is easier than direct comparison because you don't need to worry about inequality directions. Just find a similar-looking series and take the limit.

Ratio Test

Look at the ratio of consecutive terms:

lim(n→∞) |aₙ₊₁/aₙ| = L

• L < 1 → series converges absolutely
• L > 1 → series diverges
• L = 1 → test is inconclusive

This test works best when you have factorials (n!) or exponentials (3ⁿ). Those ratio out nicely. Don't bother with polynomials — the ratio will be 1 and tell you nothing.

Root Test

Take the nth root of the nth term:

lim(n→∞) √[n]{|aₙ|} = L

• L < 1 → series converges absolutely
• L > 1 → series diverges
• L = 1 → test is inconclusive

Use this when terms are raised to the nth power. (aₙ)ⁿ screams "use the root test."

Alternating Series Test

For series with alternating signs:

• The terms must decrease in absolute value
• lim(n→∞) |aₙ| = 0

If both conditions hold, the series converges. The alternating series estimation theorem lets you bound the error — the remainder is smaller than the first omitted term.

Series You Need to Memorize

Geometric Series

Σ arⁿ converges if |r| < 1 and diverges if |r| ≥ 1. The sum is a/(1-r). This is the simplest and most common series you'll encounter.

P-Series

Σ 1/nᵖ converges if p > 1 and diverges if p ≤ 1. The harmonic series (p = 1) diverges, but barely — it grows like ln(n). This is your comparison standard for anything with 1/n.

Quick Reference: Which Test When?

How to Actually Solve These Problems

Here's the process that works:

1. Calculate the nth term limit. If it's not zero, you're done — it diverges.
2. Identify what the series looks like. Geometric? P-series? Alternating?
3. Apply the matching test. Don't guess — pick the right one.
4. If you get L = 1, try a different test.

Example Walkthrough

Test Σ n/(2n+1) from n=1 to ∞.

Step 1: lim n→∞ n/(2n+1) = 1/2. Not zero. Series diverges.

Done. No further testing needed.

Another Example

Test Σ (3ⁿ)/(n!) from n=1 to ∞.

Step 1: nth term limit is 0. Keep going.

Step 2: Factorials and exponentials. Use ratio test.

Step 3: aₙ₊₁/aₙ = (3ⁿ⁺¹/(n+1)!)/(3ⁿ/n!) = 3/(n+1)

Step 4: lim n→∞ 3/(n+1) = 0 < 1. Series converges.

Absolute vs Conditional Convergence

If Σ|aₙ| converges, the series converges absolutely. Absolute convergence is stronger — it means the series converges no matter how you arrange the terms.

If Σaₙ converges but Σ|aₙ| diverges, you have conditional convergence. Alternating series often converge conditionally. Rearranging conditionally convergent series can change the sum.

Power Series Basics

A power series is a series where terms contain powers of (x-c)ⁿ. The radius of convergence is the distance from the center where the series converges. Inside that radius, it converges absolutely. Outside, it diverges. At the endpoints, test separately.

To find the radius, apply the ratio or root test to the coefficients and solve for when L < 1.

The Bottom Line

Convergence tests are decision trees, not magic. Learn the conditions for each test. Know which series converge and which diverge. Practice identifying series type quickly. The nth term test is your first filter — use it every time. Everything else follows from there.