This is a beautiful problem from ISI MStat 2018 problem 2, which uses the cute little ideas of telescopic sum and partial fractions.
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.
\(x_{n+1} = (x_{n}^{3}+\frac{1}{n(n+1)(n+2)})^{1 / 3} \Rightarrow {x_{n+1}}^3 = x_{n}^{3}+\frac{1}{n(n+1)(n+2)}\)
\( \Rightarrow {x_{n+1}}^3 - x_{n}^{3} = \frac{1}{i(i+1)(i+2)}; x_1 = 1\).
\( \Rightarrow \sum_{i = 1}^{n-1} {x_{i+1}}^3 - x_{i}^{3} = \sum_{i = 1}^{n-1} \frac{1}{n(n+1)(n+2)} ; x_1 = 1\).
\( x_{n}^{3} - x_{1}^{3} = \sum_{i = 1}^{n-1} \frac{1}{i(i+1)(i+2)} = \sum_{i = 1}^{n-1} -\frac12\left(\underbrace{\frac1{i+1} -\frac1i}\right)+\frac12\left(\underbrace{\frac1{i+2}-\frac1{i+1}}\right)\)
\(\lim_{n \to \infty} (x_{n}^{3} - x_{1}^{3}) = \sum_{i = 1}^{\infty} \frac{1}{i(i+1)(i+2)} = \sum_{i = 1}^{\infty} -\frac12\left(\underbrace{\frac1{i+1} -\frac1i}\right)+\frac12\left(\underbrace{\frac1{i+2}-\frac1{i+1}}\right) = \frac14 \)
\( \lim_{n \to \infty} x_{n}^{3} = \frac54 \Rightarrow \lim_{n \to \infty} x_{n} = ({\frac54})^\frac13 \).
This is a beautiful problem from ISI MStat 2018 problem 2, which uses the cute little ideas of telescopic sum and partial fractions.
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.
\(x_{n+1} = (x_{n}^{3}+\frac{1}{n(n+1)(n+2)})^{1 / 3} \Rightarrow {x_{n+1}}^3 = x_{n}^{3}+\frac{1}{n(n+1)(n+2)}\)
\( \Rightarrow {x_{n+1}}^3 - x_{n}^{3} = \frac{1}{i(i+1)(i+2)}; x_1 = 1\).
\( \Rightarrow \sum_{i = 1}^{n-1} {x_{i+1}}^3 - x_{i}^{3} = \sum_{i = 1}^{n-1} \frac{1}{n(n+1)(n+2)} ; x_1 = 1\).
\( x_{n}^{3} - x_{1}^{3} = \sum_{i = 1}^{n-1} \frac{1}{i(i+1)(i+2)} = \sum_{i = 1}^{n-1} -\frac12\left(\underbrace{\frac1{i+1} -\frac1i}\right)+\frac12\left(\underbrace{\frac1{i+2}-\frac1{i+1}}\right)\)
\(\lim_{n \to \infty} (x_{n}^{3} - x_{1}^{3}) = \sum_{i = 1}^{\infty} \frac{1}{i(i+1)(i+2)} = \sum_{i = 1}^{\infty} -\frac12\left(\underbrace{\frac1{i+1} -\frac1i}\right)+\frac12\left(\underbrace{\frac1{i+2}-\frac1{i+1}}\right) = \frac14 \)
\( \lim_{n \to \infty} x_{n}^{3} = \frac54 \Rightarrow \lim_{n \to \infty} x_{n} = ({\frac54})^\frac13 \).
Thank you! Really helpful ?
please check if the ans is (7/4)^1/3 . I think there is minus missing in the 4th line .