What is the difference between absolute and conditional convergence?

Excellent question — this distinction is subtle but fundamental in analysis. **Absolute convergence** means $\sum |a_n| < \infty$. This is a very strong form of convergence: the series converges regardless of the order of terms. If you rearrange an absolutely convergent series, the sum stays the same. **Conditional convergence** means $\sum a_n$ converges but $\sum |a_n|$ diverges. The classic example is the alternating harmonic series: $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac12 + \frac13 - \frac14 + \cdots = \ln 2$$ Here $\sum \frac{1}{n}$ diverges (harmonic series), but the alternating version converges by the Alternating Series Test. The **Riemann rearrangement theorem** states that a conditionally convergent series can be rearranged to converge to any real number, or even to diverge. Why? Because the positive terms alone diverge to $+\infty$ and the negative terms alone diverge to $-\infty$. By carefully reordering, you can achieve any limit. For absolute convergence, the positive and negative parts each converge individually, so rearrangement never changes the sum.