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W08 - Examples

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Simple divergence test

Simple divergence test: examples

Consider:

  • This diverges by the SDT because and not .

Consider:

  • This diverges by the SDT because .
  • We can say the terms “converge to the pattern ,” but that is not a limit value.

Positive series

p-series examples

By finding and applying the -series convergence properties:

We see that converges: so

But diverges: so

Integral test - pushing the envelope of convergence

Does converge?

Does converge?

Notice that grows very slowly with , so is just a little smaller than for large , and similarly is just a little smaller still.

Solution

The two series lead to the two functions and .

Check applicability.

Clearly and are both continuous, positive, decreasing functions on .

Apply the integral test to .

Integrate :

Conclude: diverges.

Apply the integral test to .

Integrate :

Conclude: converges.

Direct comparison test: rational functions

The series converges by the DCT.

  • Choose: and
  • Check:
  • Observe: is a convergent geometric series

The series converges by the DCT.

  • Choose: and .
  • Check:
  • Observe: is a convergent -series

The series converges by the DCT.

  • Choose: and
  • Check: (notice that )
  • Observe: is a convergent -series

The series diverges by the DCT.

  • Choose: and
  • Check:
  • Observe: is a divergent -series

Limit comparison test examples

The series converges by the LCT.

  • Choose: and .
  • Compare in the limit:
  • Observe: is a convergent geometric series

The series diverges by the LCT.

  • Choose: ,
  • Compare in the limit:
  • Observe: is a divergent -series

The series converges by the LCT.

  • Choose: and
  • Compare in the limit:
  • Observe: is a converging -series

Alternating series

Alternating series test: basic illustration

(a) converges by the AST.

  • Notice that diverges as a -series with .
  • Therefore the first series converges conditionally.

(b) converges by the AST.

  • Notice the funny notation: .
  • This series converges absolutely because , which is a -series with .

Approximating pi

The Taylor series for is given by:

Use this series to approximate with an error less than .

Solution The main idea is to use and thus . Therefore:

and thus:

Write for the error of the approximation, meaning .

By the AST error formula, we have .

We desire such that . Therefore, calculate such that , and then we will know:

The general term is . Plug in in place of to find . Now solve:

We conclude that at least terms are necessary to be confident (by the error formula) that the approximation of is accurate to within .