Page 94 - Calculus for the Clueless, Calc II

Example 10—

Example 11—

Note that test 1 implies divergence in these three examples.

Example 12—

Using partial fractions

Writing out the first few terms plus the n - 1 term plus the nth term, we get

Notice all the middle terms cancel out in pairs. So only the first and last terms remain:

Again, this is one of the few sequences we can find the exact value for. (This is called a telescoping series—it
collapses like one of those toy or portable telescopes.) From this point on, for almost all of the converging
series, we will be able to tell that the series converges, but we won't be able to find its value. Later we will do
some approximating.

Example 13—

After splitting, we get two geometric series:

Theorem

Now that we have an idea about what a sequence is and what an infinite series is (hopefully a very good idea),
we would like to have some tests for when a series converges or diverges.

Test 1