Page 93 - Calculus for the Clueless, Calc II

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Using part 6 above, -1 < sin n < 1. So -1/n < (sin n)/n < 1/n. As n goes to infinity, -1/n and 1/n go to 0.
Therefore, so does (sin n)/n.

Definition 1

An increasing sequence is one where a n < a n+1 for all n.

Definition 2

A non decreasing sequence is one where a n < a n+1 for all n.

Similarly we can define decreasing and nonin-creasing.

Definition 3

A sequence is bounded if | a n | < M, some number M and all n.

Another theorem: Every bounded increasing (decreasing) sequence has a limit.

Infinite Series

I know this is getting to be a drag, but it is essential to understand the terminology. This understanding will
make the rest of the chapter much easier. I don't know why, but it really seems to.

Definition

Partial sums—Given sequence {a n}:

1st partial sum S 1 = a 1

2nd partial sum S 2 = a 1 + a 2

3rd partial sum S 3 = a 1 + a 2 + a 3

nth partial sum

The infinite series a 1 + a 2 + a 3 + ... or is said to converge to the sum S if . If S does not
exist, the series diverges.

Example 8—

.767676 ....
We can write this as an infinite series. .76 + .0076 + .000076 + .... This is a geometric series (infinite). This is
one of the few series we can find the exact sum of.

More generally, the series a + ar + ar + ar + ... converges to a/(1 - r) if | r | < 1.
3
2
Example 9—