Question

An arithmetic series is a series where there is a constant difference between successive terms; the general form for such a series is ∑n=0[infinity](a+nd) where d is the common difference. Under what circumstances does an arithmetic series converge?

Answer 1

An arithmetic series converges if the common difference is less than 1 in absolute value, causing the terms to get smaller and the sum to approach a finite value. For example, the series 3, 1, -1, -3, ... is convergent because the terms are decreasing and approaching 0 as n increases.

Step-by-step explanation:

An arithmetic series converges under certain conditions. Specifically, an arithmetic series will converge if the common difference (denoted as 'd') is less than 1 in absolute value. This means that the terms of the series are getting smaller and smaller as n (the number of terms) increases. As a result, the sum of the series will approach a finite value as n approaches infinity.

For example, the arithmetic series 3, 7, 11, 15, ... has a common difference of 4. This series is divergent because the terms are increasing as n increases. However, the arithmetic series 3, 1, -1, -3, ... has a common difference of -2. This series is convergent because the terms are decreasing and approaching 0 as n increases.