Divergence and Convergence of Arithmetic Series

Question

Can a series using the arithmetic sequence ever converge? Why?

Does it change if it's a finite series vs an infinite series?

Answer 1

An arithmetic sequence has the general form a_n = a_1 + (n-1)d. For example, suppose you are working with the sequence a_n = a_n-1 + 2, i.e., to go from one term to the next in the sequence you add 2. If you do this a finite number of times, say 10 times, you know that the final term in your sequence is your first term + 20, so the sequence converges. However, if you have an infinite sequence, there is no final term and each term becomes larger and larger, so this sequence must diverge. Be careful to distinguish a sequence from a series. A sequence is just a progression of numbers like 1,3,5,7,..., whereas a series is a sum of those terms, 1+3+5+7+...

Answer 2

A finite series always converges because summing up a finite number of finite (real numbers) cannot approach infinity and diverge since its exact value can be calculated. Now if the series is infinite, it may converge or diverge. The series ∑^inf_k=1 (a_k) converges if and only if the limit of the sequence of partial sums (sum of the first number, first two numbers, first three numbers, etc.) S₁, S₂, S₃, ... converges. Now the nth term of an arithmetic sequence is in the form a_n = a₁ + (n-1)d. Thus, the nth partial sum is Σⁿ_k=1(a₁ + (k-1)d) = Σⁿ_k=1(a₁) + d * Σⁿ_k=1(k-1) by splitting up the summation and using the sum and product rules. The first term is equal to n*a₁ since the constant is being added a total of n times, and the second term evaluates to d * (n-1)n/2 by using the triangle number formula. Combining this yields S_n = (d/2) * n² + (a₁-d/2)*n. Now the limit as n approaches ∞ of S_n is either +∞ or -∞ depending on whether d is positive or negative because d/2 is non zero since d is non zero by definition, so we only need to look at the coefficient of the highest power of n (n²) to find in what direction it goes. However, in either case, the sequence diverges, so the series must diverge.