Find the sum of the infinite geometric series given by:  .

Before we do anything, we'd better make sure our series is convergent. Otherwise, we can't find an infinite sum at all. What's our common ratio?

Yep, |r| < 1 so the series converges. Now we can nab our infinite geometric series formula.

Substitute for a₁ and r in the formula and watch the magic unfold.

S = 4.5

Find the sum of the infinite geometric series given by:  .

Let's hunt down that common ratio. It's just the number in parentheses.

r = 3

This series diverges because |r| ≥ 1. Therefore the sum is infinite. Hey, that means we're done!

Find the sixth partial sum of the geometric series given by: .

Use the formula for the partial sum of a geometric series. You've got it printed out on a little card in your wallet, right?

Substitute the values you know into the formula.

Now slog through the actual math and simplify everything as much as you can.

Find the sum: .

Use the formula for the partial sum of a geometric series. It's our best bud.

Now pop in the first term (a₁) and the common ratio (r). Use n = 3, since we're after the 3rd partial sum.

Rock those fractions.