Convergence Test for Alternating Series and When to Use It

In summary, the conversation discusses the convergence of the series Ʃ cos(k*pi)/k from 1 to infinity and the proper use of the alternating series test. It is determined that the series converges by the direct comparison test, and that the limit of (4/8)^k as k approaches infinity is 0. It is also mentioned that for a series involving sine or cosine over a variable, the comparison test is (-1)^k, while for sine or cosine squared, the comparison is 1.
  • #1
ichilouch
9
0

Homework Statement


Ʃ cos(k*pi)/k from 1 to infinity.
This is a test for convergence.

and when is the proper time to use the alternating series test
like using it on (-1)k(4k/8k) would result to divergence
since lim of (4k/8k) is infinity and not 0 but the function is really
convergent by the geometric series?

Homework Equations





The Attempt at a Solution


Is it right to do this:

for k is odd cos is negative and for k is even cos is positive
then
cos(k*pi)\k < (-1)k/k
and by alternating series test ;
(-1)k*(1/k) since 1/k is decreasing and lim as 1/k approaches infinity is 0 then
(-1)k1/k converges thus cos(k*pi)/k converges by direct comparison test.
[
 
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  • #2
That is perfectly OK for the alternating 1/k-series! :smile:

"since lim of (4^k/8^k) is infinity"
Is it?
 
  • #3
Thanks for the reply
So rechecking limit of (4/8)^k as k approaches infinity is 0
Then if it is sine or cosine over a variable, the comparison test for it is (-1)^k?
and if it is sine or cosine squared, the comparison is 1 only?
 

1. What is a convergence test?

A convergence test is a method used to determine whether or not a mathematical series will converge (approach a finite value) or diverge (have no finite limit). It is commonly used in calculus and other branches of mathematics to evaluate the behavior of infinite series.

2. Why is it important to use a convergence test?

Using a convergence test is important because it allows us to determine the behavior of a series, which can help us make predictions and draw conclusions in various mathematical applications. It also helps us determine if a series is convergent or divergent, which is essential for solving many problems in mathematics.

3. What are some common types of convergence tests?

Some common types of convergence tests include the integral test, the comparison test, the ratio test, and the root test. Other tests include the alternating series test, the limit comparison test, and the condensation test.

4. How do I know which convergence test to use?

There is no one-size-fits-all answer for which convergence test to use in a given situation. It often depends on the specific series and its properties. It is important to understand the different convergence tests and their conditions for use, and then apply them to the series in question to determine the best fit.

5. What are some tips for using convergence tests effectively?

Some tips for using convergence tests effectively include being familiar with the different tests and their conditions for use, carefully checking the conditions of each test before applying it, and practicing with a variety of examples to become more comfortable with using them. It is also important to understand the concept of convergence and divergence in order to interpret the results of the tests correctly.

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