Look at the arithmetic series
where ai = 4 + 3(i – 1)
Does this series converge or diverge?
Finding a formula for the partial sum Sn would be a bit annoying, but thankfully we don't need to bother with that. We know that Sn is the sum of the first n terms of the series:
Sn = (4 + (1 – 1)3) + (4 + (2 – 1)3) + ... + (4 + (n – 1)3).
Each term is 4 plus something non-negative:
Since each of the n terms we're adding up is at least 4, the partial sum is at least n times 4:
Sn ≥ 4n.
From here it should be easy to see that as n approaches ∞, so does Sn (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.
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