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W07 - Examples

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Sequences

Geometric sequence: revealing the format

Find and and (written in the geometric sequence format) for the following geometric sequences:

(a) (b) (c)

Solution (a) Plug in to obtain . Notice that and so therefore . Then the ‘general term’ is .

(b) Rewrite the fraction:

Plug that in and observe . From this format we can read off and .

(c) Rewrite:

From this format we can read off and .

L’Hopital’s Rule for sequence limits

(a) What is the limit of ? (b) What is the limit of ? (c) What is the limit of ?

Solution (a) Identify indeterminate form . Change from to and apply L’Hopital:

(b) Identify indeterminate form . Change from to and apply L’Hopital:

(c) Identify form and rewrite as :

Change from to and apply L’Hopital:

Simplify:

Consider the limit:

Squeeze theorem

Use the squeeze theorem to show that as .

Solution We will squeeze the given general term above and below a sequence that we must devise:

We need to satisfy and . Let us study .

Now for the trick. Collect factors in the middle bunch:

Each factor in the middle bunch is so the entire middle bunch is . Therefore:

Now we can easily see that as , so we set and we are done.

Monotonicity

Show that converges.

Solution

Observe that for all .

Because , we know .

Therefore

Change to and show is decreasing.

New formula: considered as a differentiable function.

Take derivative to show decreasing.

Derivative of :

Simplify:

Denominator is . Numerator is . So and is monotone decreasing.

Therefore is monotone decreasing as .

Series

Geometric series - total sum and partial sums

The geometric series total sum can be calculated using a “shift technique” as follows:

  1. Compare and :
  2. Subtract second line from first line, many cancellations:
  3. Solve to find :

Note: this calculation assumes that exists, i.e. that the series converges.

The geometric series partial sums can be calculated similarly, as follows:

  1. Compare and :
  2. Subtract second line from first line, many cancellations:
  3. Solve to find :
  • The last formula is revealing in its own way. Here is what it means in terms of terms: