An infinite series is a sum of infinitely many terms, typically represented in the form of a sequence. It is called "infinite" because there is no specific number of terms in the series.
To prove that an infinite series converges, you can use various methods such as the comparison test, the ratio test, or the integral test. These methods involve comparing the given series to a known convergent or divergent series and determining the convergence or divergence of the given series based on the comparison.
An infinite series diverges when the sum of its terms approaches infinity, meaning that the series does not have a finite limit. This can happen when the terms of the series do not approach zero or when they alternate in sign indefinitely.
Yes, an infinite series can converge to a negative number. For example, the alternating harmonic series (-1 + 1/2 - 1/3 + 1/4 - 1/5 + ...) converges to ln(2), which is a negative number.
The convergence of an infinite series is dependent on the convergence of its terms. If the terms of a series do not approach zero, the series will diverge. On the other hand, if the terms of a series approach zero, the series may still diverge or converge, and further tests are needed to determine its convergence.