SeriesMedium

Root test — series convergence

Determine whether the series Σ_{n=1}^{∞} (n/(2n+1))^n converges using the root test.

Given data

Σ (n/(2n+1))^n

Steps

0 / 3

Apply the root test: calculate the limit of ⁿ√(a_n).
Write your working — the numerical result must appear in the text (logica)
Calculate the limit: lim_{n→∞} n/(2n+1).
Compare with 1: ρ = 1/2 < 1. What is the conclusion?

Full worked solution

Apply the root test: calculate the limit of ⁿ√(a_n).
$\lim_{n\to\infty} \sqrt[n]{a_n} = \lim_{n\to\infty} \dfrac{n}{2n+1}$
$a_n = \left(\frac{n}{2n+1}\right)^n$ , so $\sqrt[n]{a_n} = \frac{n}{2n+1}$ . The root test studies $\rho = \lim_{n\to\infty} \sqrt[n]{|a_n|}$ .
Calculate the limit: lim_{n→∞} n/(2n+1).
$\lim_{n\to\infty} \dfrac{n}{2n+1} = \dfrac{1}{2}$
$\rho = \lim_{n\to\infty} \frac{n}{2n+1} = \lim_{n\to\infty} \frac{1}{2 + 1/n} = \frac{1}{2}$ . Dividing numerator and denominator by $n$ reveals the limit.
Compare with 1: ρ = 1/2 < 1. What is the conclusion?
$\rho = \dfrac{1}{2} < 1 \Rightarrow \text{converge}$
$\rho = \frac{1}{2} < 1$ → the series converges (by the root test). When $\rho < 1$ , the series converges absolutely.

Result:The series converges (ρ = 1/2 < 1).

Keep studying