MENU

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:

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.

Make it rain.

The who, what, where, when, and why of all your favorite quotes.

Go behind the scenes on all your favorite films.

You've been inactive for a while, logging you out in a few seconds...