1. The problem statement, all variables and given/known data Prove that if the infinite series a_1 + a_2 + ... + a_v converges to a value A and s_n = a_1 + a_2 + a_3 + ... + a_n, then the sequence: (s_1 + s_2 + ... + s_N)/N also converges, and has the limit A. 2. Relevant equations 3. The attempt at a solution Since s_n represents the n'th partial sum, then s_1 + s_2 + s_3 + ... s_N = a_1 + (a_1 + a_2) + (a_1 + a_2 + a_3) + ...+ (a_1 + ... + a_N) = N*a_1 + (N-1)*a_2 + (N-2)a_3 + ... + a_N So our sequence looks like (N*a_1 + (N-1)*a_2 + (N-2)a_3 + ... + a_N)/N Notice that the terms in front of the a_i form a monotonic decreasing sequence which converges to 0: N/N = 1, (N-1)/N, (N-2)/N, ... , 1/N So now I use Abel's Test: "Let a_1 + a_2 + ... be an infinite series whose partial sums are bounded independent of n. Let p_1, p_2, ... be a sequence of positive numbers decreasing monotonically to the value 0, then the infinite series p_1*a_1 + p_2*a_2 + ... converges" Since my original series converges, its partial sums are bounded, and the terms in front of the a_i form a monotonically decreasing sequence going to 0, just like I said, and so the infinite series converges... I'm just having trouble proving that it converges to the value A. I have no idea where to start.