Exercise 791Calculus › Single-variable calculus › Series › Convergence testsL1Added 2 Nov 2026

The root test

#root test#Cauchy#exponential limits

Statement

Decide the nature of
$\sum\left(\frac n{2n+1}\right)^n,\qquad\sum\frac1{3^n}\left(1+\frac1n\right)^{n^2},\qquad\sum\left(1-\frac1n\right)^{n^2}.$

Hint

If $\sqrt[n]{u_n}\to L$ , the series converges for $L<1$ and diverges for $L>1$ .

Show solutionHide solution

Solution

1. $\sqrt[n]{u_n}=\frac n{2n+1}\to\frac12<1$ : converges.

2. $\sqrt[n]{u_n}=\frac13\left(1+\frac1n\right)^n\to\frac e3\approx0.906<1$ : converges.

3. $\sqrt[n]{u_n}=\left(1-\frac1n\right)^n\to\frac1e<1$ : converges.

Why the root test? The exponent $n^2$ or $n$ makes the ratio test awkward but the $n$ -th root easy.

Remark. The root test is more powerful than the ratio test: whenever the ratio test gives an answer, so does the root test (the converse is false).

The root test

Statement

Solution

More problems in Convergence tests