Series Convergence Tests: Comparison, Ratio, Root, Integral

When you meet an infinite series such as Σ an, the first question is always the same: does it add up to a finite number, or does it grow without bound? The comparison test, the ratio test, the root test and the integral test are the four standard tools for answering that question when the terms are positive. Each test examines the terms in a different way, and knowing all four lets you choose the fastest route for any given series.

These tests matter because infinite series are everywhere: calculators approximate sin x and e^x with Taylor series, physics formulas are often infinite sums, and numerical methods depend on series that actually settle down. If you cannot decide whether a series converges, you cannot trust the number it claims to produce.

Key Concepts

• Convergence of a series: Σ an converges when its partial sums SN = a1 + a2 + … + aN approach a finite limit as N grows.
• Comparison test: if 0 ≤ an ≤ bn and Σ bn converges, then Σ an converges; if 0 ≤ bn ≤ an and Σ bn diverges, then Σ an diverges.
• Limit comparison test: if an, bn > 0 and the limit of an/bn is a finite positive number L, then Σ an and Σ bn either both converge or both diverge.
• Ratio test: compute L = limit of |an+1/an|. If L < 1 the series converges absolutely, if L > 1 it diverges, and if L = 1 the test says nothing.
• Root test: compute L = limit of |an|1/n. The conclusions are the same as for the ratio test, and L = 1 is again inconclusive.
• Integral test: if f is positive, decreasing and continuous on [1, ∞) with f(n) = an, then Σ an and the integral from 1 to ∞ of f(x) dx either both converge or both diverge.
• p-series: Σ 1/np converges exactly when p > 1 and diverges for p ≤ 1. It is the benchmark series used in comparison tests.

Formulas

• Ratio test: L = lim |an+1/an|. Works best on factorials and exponentials such as n!/3n. Requires the limit to exist; inconclusive at L = 1.
• Root test: L = lim |an|1/n. Works best on n-th powers such as (2n/(3n+1))n. Inconclusive at L = 1.
• Integral test: the integral from 1 to ∞ of dx/xp equals 1/(p−1) for p > 1 and diverges for p ≤ 1. Only valid when f is decreasing, so check the derivative first.
• Limit comparison: a finite positive limit of an/bn means the two series share the same fate. Ideal for rational expressions in n.

Worked Example

Decide whether Σ n/2n converges, starting at n = 1.

1. The terms mix a power of n with an exponential, so the ratio test is the natural choice. Set an = n/2n.
2. Form the ratio: |an+1/an| = ((n+1)/2n+1) ÷ (n/2n) = (n+1)/(2n).
3. Take the limit as n → ∞: (n+1)/(2n) = 1/2 + 1/(2n), which tends to 1/2.
4. Since L = 1/2 < 1, the series converges absolutely by the ratio test.

Its exact sum is in fact 2, which you can find by differentiating the geometric series, but the test alone settles the question of convergence.

Common Misconceptions

• "L = 1 in the ratio test means divergence." It means nothing at all. Both Σ 1/n² (convergent) and Σ 1/n (divergent) give L = 1.
• "If an → 0 then Σ an converges." Terms tending to zero is necessary but not sufficient; the harmonic series Σ 1/n diverges even though 1/n → 0.
• "The integral test works for any positive terms." The function must also be decreasing; an oscillating positive function can make the integral misleading.

Real-World Applications

• Verifying that the Taylor series used by calculators and software libraries for sin x, ex and ln(1+x) actually converge.
• Proving convergence of Fourier series in audio compression and signal processing.
• Analysing algorithms in computer science whose total cost is an infinite-looking sum, such as the expected cost of randomized algorithms.

Use the comparison test when a series resembles a known benchmark, the ratio test for factorials and exponentials, the root test for n-th powers, and the integral test to settle p-series. Always check first that the terms tend to zero, and remember that a ratio-test value of exactly 1 tells you nothing and you must try another test.