Root Test

Core Idea

The Root Test examines L = lim(n->infinity) |a_n|^(1/n). If L < 1, the series converges absolutely. If L > 1, it diverges. If L = 1, inconclusive. The root test is most useful when a_n involves an nth power, such as (expression)ⁿ, where the nth root simplifies cleanly. It is equivalent in power to the ratio test but sometimes easier to apply.

How It's Best Learned

Apply to series of the form (f(n))ⁿ. Compare with the ratio test on the same series to see which is more convenient. Practice computing nth roots using properties of limits and logarithms.

Common Misconceptions

• Trying to use the root test on series with factorials (the ratio test is usually better for those).
• Confusing |a_n|^(1/n) with |a_n^1/n| (they are the same, but the computation can be confusing).
• Drawing a conclusion when L = 1.

Explainer

The Root Test is a convergence test for infinite series that works by comparing a series to a geometric series — one of the few series types whose convergence you can determine completely. Recall that a geometric series Σrⁿ converges if and only if |r| < 1. The Root Test asks: does the n-th term of your series behave like rⁿ for some r, and if so, what is r? It answers this by looking at the n-th root of the n-th term.

The key quantity is L = lim_{n→∞} |aₙ|^{1/n}. If this limit exists and equals L, then for large n, |aₙ| behaves like Lⁿ. If L < 1, the terms shrink like a convergent geometric series, so the series converges absolutely. If L > 1, the terms grow, so the series diverges (the terms don't even go to zero). If L = 1, the test gives no information — both convergent and divergent series can have L = 1, so you need a different test.

The test shines on series where the n-th term is itself an n-th power, because the n-th root then simplifies cleanly. For example, consider Σ(2n/(3n+1))ⁿ. Taking the n-th root gives |aₙ|^{1/n} = 2n/(3n+1) → 2/3 as n → ∞. Since 2/3 < 1, the series converges absolutely. By contrast, the Ratio Test on this series would require computing ratios of (2n/(3n+1))ⁿ and (2(n+1)/(3(n+1)+1))ⁿ⁺¹, which is messier. Whenever you see an expression raised to the n-th power as the n-th term, the Root Test is usually your first choice.

Compare the Root Test to the Ratio Test, which you already know: both tests ultimately compare the series to a geometric series, and they are theoretically equivalent in power (if one gives L, the other gives the same L). However, the Root Test handles n-th powers more cleanly, while the Ratio Test handles factorials more cleanly (because factorial ratios simplify to single terms). When both are applicable, they give the same conclusion. The L = 1 inconclusive case is unavoidable — this is where series like Σ1/n (diverges) and Σ1/n² (converges) both land, and p-series or comparison tests are needed instead.