Final answer:
If the limit of a partial sum exists and is real (i.e. if sn converges to s), we say the series converges. The series has a finite sum. On the other hand, if the partial sum does not approach a specific real number, we say the series diverges.
Step-by-step explanation:
If the limit of a partial sum exists and is real (i.e. if sn converges to s), we say the series ∑ k=m to infinity of ak converges. In other words, the series has a finite sum. This means that as we add more terms to the series, the value of the series approaches a specific real number.
For example, let's consider the series ∑ n=1 to infinity of 1/n^2. This is known as the harmonic series. It converges to a specific real number, which is approximately 1.645. Therefore, the series converges.
On the other hand, if the partial sum does not approach a specific real number, we say the series diverges. This means that as we add more terms to the series, the value of the series becomes increasingly larger or smaller without approaching a finite sum.
For example, consider the series ∑ n=1 to infinity of (-1)^n. This is known as the alternating series. The partial sums of this series alternate between 1 and -1, never approaching a specific real number. Therefore, the series diverges.