This is a beautiful problem from ISI MStat 2018 problem 2, which uses the cute little ideas of telescopic sum and partial fractions.

Problem

Let \(\{x_{n}\}_{n \geq 1}\) be a sequence defined by \(x_{1}=1\) and
$$
x_{n+1}=\left(x_{n}^{3}+\frac{1}{n(n+1)(n+2)}\right)^{1 / 3}, \quad n \geq 1
$$
Show that \(\{x_{n}\}_{n \geq 1}\) converges and find its limit.

Prerequisities

  • Limit of a Sequence
  • Partial Fraction \( \frac{1}{n(n+1)(n+2)} = \frac1{n(n+1)(n+2)}=\frac12\cdot\frac1n-\frac1{n+1}+\frac12\cdot\frac1{n+2} = -\frac12\left(\underbrace{\frac1{n+1} -\frac1n}\right)+\frac12\left(\underbrace{\frac1{n+2}-\frac1{n+1}}\right)\)
  • Telescopic Sum \( \sum_{i = 1}^{\infty} \left(\underbrace{\frac1{i+1} -\frac1i}\right) = \lim_{n \to \infty} \frac1n – 1 = -1 \)

Solution

\(x_{n+1} = (x_{n}^{3}+\frac{1}{n(n+1)(n+2)})^{1 / 3} \Rightarrow {x_{n+1}}^3 = x_{n}^{3}+\frac{1}{n(n+1)(n+2)}\)

\( \Rightarrow {x_{n+1}}^3 – x_{n}^{3} = \frac{1}{i(i+1)(i+2)}; x_1 = 1\).

\( \Rightarrow \sum_{i = 1}^{n-1} {x_{i+1}}^3 – x_{i}^{3} = \sum_{i = 1}^{n-1} \frac{1}{n(n+1)(n+2)} ; x_1 = 1\).

\( x_{n}^{3} – x_{1}^{3} = \sum_{i = 1}^{n-1} \frac{1}{i(i+1)(i+2)} = \sum_{i = 1}^{n-1} -\frac12\left(\underbrace{\frac1{i+1} -\frac1i}\right)+\frac12\left(\underbrace{\frac1{i+2}-\frac1{i+1}}\right)\)

\(\lim_{n \to \infty} (x_{n}^{3} – x_{1}^{3}) = \sum_{i = 1}^{\infty} \frac{1}{i(i+1)(i+2)} = \sum_{i = 1}^{\infty} -\frac12\left(\underbrace{\frac1{i+1} -\frac1i}\right)+\frac12\left(\underbrace{\frac1{i+2}-\frac1{i+1}}\right) = \frac14 \)

\( \lim_{n \to \infty} x_{n}^{3} = \frac54 \Rightarrow \lim_{n \to \infty} x_{n} = ({\frac54})^\frac13 \).