Arithmetic Series Examples

Look at the arithmetic series

where a_i = 4 + 3(i – 1)

Does this series converge or diverge?

Finding a formula for the partial sum S_n would be a bit annoying, but thankfully we don't need to bother with that. We know that S_n is the sum of the first n terms of the series:

S_n = (4 + (1 – 1)3) + (4 + (2 – 1)3) + ... + (4 + (n – 1)3).

Each term is 4 plus something non-negative:

S_n = (4 + (1 – 1)3) + (4 + (2 – 1)3) + ... + (4 + (n – 1)3).

Since each of the n terms we're adding up is at least 4, the partial sum is at least n times 4:

S_n ≥ 4n.

From here it should be easy to see that as n approaches ∞, so does S_n (if this isn't easy to see, go review limits). Since

is not a finite number, the sequence of partial sums diverges. This means the original arithmetic series diverges.