Useful Limits

Exponentials

Powers and Roots

Properties of Limits

The limit of a sum is the sum of the limits:

The limit of a product is the product of the limits:

The limit of a quotient is the quotient of the limits:

Squeeze Theorem:

AKA the "Flyswatter Theorem". Why?

Ratios of Polynomials

The limiting behavior of the ratio of two polynomials depends on the degree of the polynomials. The polynomial of higher degree "wins". If their degrees are the same, then the limit is the ratio of the leading coefficients.

Useful Inequalities

Always

Eventually

1. 
2. for any
3. (!) Remember this when using the Test for Divergence

New Series From Old

Multiply by a constant:

Before

After

Substitute an expression for x:

Before

After

Multiply by a power of x:

Before

After

Integrate:

Before

After

Differentiate:

Before

After

Convergence Tests

Tips for Series

• Often the hardest part of showing convergence or divergence of a series is the indecision: What do I believe it does? After all, you'll have a tough time showing a series converges if it doesn't!
• The limits listed in UsefulLimits can help a lot with the Test for Divergence. Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.
• Many limits boil down to "look like" ratios of polynomials after stripping out trig functions using the Useful Inequalities for trig functions.
• The eventual behavior that for any leads to the peculiar rule of thumb that in lots of ratios ln(n) "looks like" 1 since any positive power of n will dominate it:
  • informally, "looks like" so converges
  • more carefully, (eventually),
    so,
    which is a convergent p-series.