A convergent infinite series is a series where the sum of its terms approaches a finite value as you add more and more terms. In other words, it settles down to a specific number. Think of it like walking closer and closer to a specific spot; eventually, you'll get there!
A divergent infinite series, on the other hand, is a series where the sum of its terms does not approach a finite value. Instead, it either goes to infinity (positive or negative) or oscillates without settling on any specific value. It's like walking without a destination; you just keep wandering!
| Feature | Convergent Series | Divergent Series |
|---|---|---|
| Definition | Sum approaches a finite value. | Sum does not approach a finite value. |
| Limit of Partial Sums | Exists and is finite. | Does not exist or is infinite. |
| Behavior | Settles down to a specific number. | Goes to infinity or oscillates. |
| Example | $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = 2$ | $1 + 2 + 3 + 4 + ... = \infty$ |
| Mathematical Notation | $\sum_{n=1}^{\infty} a_n = L$ (where L is a finite number) | $\sum_{n=1}^{\infty} a_n = \infty$ or oscillates |
| Common Tests | Ratio Test, Root Test, Comparison Test | Divergence Test, Integral Test |
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