## The Convergence of (n^3 + 1)^(1/3) - n

This article delves into the convergence of the sequence defined by the expression **(n^3 + 1)^(1/3) - n**, where 'n' represents a positive integer. We will explore its behavior as 'n' approaches infinity.

### Understanding the Expression

The expression **(n^3 + 1)^(1/3) - n** represents the difference between the cube root of (n^3 + 1) and the integer 'n'. To understand its convergence, we can analyze the expression further.

### Simplifying the Expression

We can simplify the expression by using the algebraic identity: **a^3 - b^3 = (a - b)(a^2 + ab + b^2)**

Let's set:

**a = (n^3 + 1)^(1/3)****b = n**

Applying the identity, we get:

**(n^3 + 1)^(1/3) - n = ((n^3 + 1)^(1/3) - n) * (((n^3 + 1)^(2/3) + n(n^3 + 1)^(1/3) + n^2)) / (((n^3 + 1)^(2/3) + n(n^3 + 1)^(1/3) + n^2))**

This simplifies to:

**1 / ((n^3 + 1)^(2/3) + n(n^3 + 1)^(1/3) + n^2)**

### Analyzing the Simplified Expression

As 'n' approaches infinity, the denominator of this expression grows much faster than the numerator. This is because the dominant terms in the denominator are **n^2** and **n(n^3 + 1)^(1/3)**, both of which increase at a faster rate than the constant numerator '1'.

Therefore, as 'n' approaches infinity, the simplified expression converges to **0**.

### Conclusion

The sequence defined by **(n^3 + 1)^(1/3) - n** converges to **0** as 'n' approaches infinity. This convergence is driven by the fact that the denominator grows much faster than the numerator, resulting in a value approaching zero.