Important points about the alternating series test, Mathematics

Assignment Help:

Important Points About the Alternating Series Test

There are a several things to note about this test.  Very first, unlike the Integral Test and the Comparison or Limit Comparison Test, this test will just only tell us while a series converges and not if a series will diverge.

After that 2nd, in the second state all that we need to require is that the series terms, bn will be eventually decreasing.  It is probable for the first few terms of a series to raise and still have the test be valid.  All that is needed is that eventually we will have bn ≥ bn+1 for all n later than some point.

 To observe why this is consider the subsequent series,

1288_Important Points About the Alternating Series Test 1.png

Let us assume that for 1<n<N {bn} is not decreasing and that for n > N+1 {bn} is decreasing. The series can be written like:

1093_Important Points About the Alternating Series Test 2.png

Here the first series is a finite sum there is no matter how large N is of finite terms and thus we can calculate its value and it will be finite.  The convergence of this series will depend upon only on the convergence of the second that is infinite series.  If the 2nd series has a finite value as compared to the sum of two finite values is as well finite and thus the original series will converge to a finite value.  Alternatively if the second series is divergent either as its value is infinite or it doesn't have a value after that adding a finite number onto this will not change that fact and thus the original series will be divergent.

The point of all this is that we do not need to require that the series terms be decreasing for each n.  We just only need to require that the series terms will eventually be decreasing as we can all time strip out the first few terms that aren't in fact decreasing and look only at the terms which are actually decreasing.

 Note: in reality, we don't actually strip out the terms which aren't decreasing.  All we do is test that eventually the series terms are decreasing and after that apply the test.


Related Discussions:- Important points about the alternating series test

Partial Differential Equations Walter A Strauss, Find the full fourier Seri...

Find the full fourier Series of e^x on (-l,l)in its real and complex forms. (hint:it is convenient to find the complex form first)

Percentage, of all those survey 390 were under 18 years of age if 20%were 1...

of all those survey 390 were under 18 years of age if 20%were 18, how many responded to the survey

Fractions, If i worked 7 1/3 hours and planted 11 trees how many hours did ...

If i worked 7 1/3 hours and planted 11 trees how many hours did it take to plant each tree?

Naming fractions greater than 1, the 10 miles assigned to the chess club st...

the 10 miles assigned to the chess club start at the 10 mile point and go to the 20 mile point when the chess club members have cleaned 5/8 of their 10 mile section between which m

Determine the volume of the box, Safe deposit boxes are rented at the bank....

Safe deposit boxes are rented at the bank. The dimensions of a box are (22x5x5) in. Determine the volume of the box? a. 220 in 3 b. 550 in 3 c. 490 in 3 d. 360 in 3

Fractions, how to add a fraction with an uncommon denomoninator

how to add a fraction with an uncommon denomoninator

What is the cost per ounce of detergent, A 64-ounce bottle of detergent cos...

A 64-ounce bottle of detergent costs $3.20. What is the cost per ounce of detergent? To ?nd out the cost per ounce, divide the cost through the number of ounces; $3.20 ÷ 64 =

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd