Convergence Tests

Divergence test

Direct Comparison Test

Let .

Proof Hint

• Note that and are increasing
• Show that converges to its supremum which is an upper bound of

Example

Proving the convergence of , by using for all .

Limit Comparison Test

Let and .

Proof Hint

Only possibilities are .

For :

• Consider limit definition with
• Direct comparison test can be used for the 2 set of inequalities

For :

• Consider limit definition with
• Direct comparison test can be used now

For :

• Consider limit definition with
• Direct comparison test can be used now

Integral Test

Let , decreasing and integrable on for all . Then:

Proof Hint

As is decreasing, it is apparent that it is integrable.

Make use of this inequality:

For :

• Note that is increasing
• Show that is bounded above by

For :

• Define
• Note that is increasing
• Note that
• Show that is bounded above by

Note

Ratio Test

Let and .

Proof Hint

• Consider the limit definition with:
  • For the case:
  • For the case:
• Show that:
• Recursively simplify the inequality to reach which is a constant
• Use is converging iff

Root Test

Let is a sequence and .

Proof Hint

Consider the limit definition with:

• For :
• For :
• For :

Dirichlet’s test

Let:

• is a decreasing sequence, converging to and
• is a sequence and
• is bounded