Series Exercises

Answer

The first 5 partial sums are

There are several things you could notice. Here are some of them:

• The numbers seem to come in pairs, one negative and then one positive. We get-2, + 2, -4, + 4.
• The partial sums alternate between positive and negative.
• The partial sums go up, then down, then up.

Answer

(a)

Writing all the partial sums with a denominator of 32, we get

(b) The graph looks like this:

(c) The partial sums go down, then up, then down, then up, while getting closer together. It looks like they're converging to something.

(d) We know that if the sequence of partial sums converges then the series converges. Since it looks like the sequence of partial sums is converging to something, we would guess that the series converges.

Answer

This is an alternating series. The magnitudes of the terms are strictly decreasing:

The absolute values of the terms converge to zero:

Thus the AST says this series converges.

Answer

This is an alternating series. The magnitudes of the terms are strictly decreasing:

The absolute values converge to zero:

Thus the AST says this series converges.

Answer

This is not an alternating series. Since (2n + 1) is always odd, the quantity

is always negative. We can't use the AST here.

Answer

This is an alternating series. Since

0.1 < 1

we get smaller numbers as we take higher powers. This gives us both conditions:

|a_n| = 0.1ⁿ > 0.1^{n + 1} = |a_{n + 1}|

The AST says this series converges.

Answer

We can't use the AST here because the magnitudes of the terms don't converge to zero.

diverges.

Answer

(a) Look at some partial sums:

The partial sums jump down, then up, with S₉ being an up-jump.

To get to the next partial sum we jump down:

and in so doing, we overshoot the line:

This means when we jumped down by , it was too much. The real sum of the series, L, must be within  of S₉.

In symbols,

(b) We have

Since the last term in this partial sum has a negative sign, we jumped down to get to S₉₈. To get to S₉₉ we jump up by , but this is too big a jump. The value we want is within  of S₉₈. In symbols,