Integral test- harmonic series, Mathematics

Assignment Help:

Integral Test- Harmonic Series

In harmonic series discussion we said that the harmonic series was a divergent series.  It is now time to demonstrate that statement.  This proof will as well get us started on the way to our next test for convergence that we will be looking at.

 Thus, we will be trying to verify that the harmonic series,

1144_Integral Test- Harmonic Series 1.png

We'll initiate this off by looking at an apparently not related problem.  Let us start off by asking what the area under f (x) =1/x on the interval [1, ∞).  From the segment on Improper Integrals we come to know that this is,

1 1/x (dx) = ∞

and thus we called this integral divergent.

Thus, just how does that help us to demonstrate that the harmonic series diverges? Well, remind that we can all time estimate the area by breaking up the interval into segments and then drawing in rectangles and using the sum of the area all of the rectangles as an approximation of the actual area. Let us do that for this problem also and see what we acquire.

We will divide the interval into subintervals of width 1 and we will take the function value at the left endpoint as the height of the rectangle. The diagram below illustrates the first few rectangles for this area.

905_Integral Test- Harmonic Series 2.png

Thus, the area under the curve is approximately,

1735_Integral Test- Harmonic Series 3.png

Now note some things about this approximation.  1st, each of the rectangles overestimates the actual area and 2nd the formula for the area is precisely the harmonic series!


Related Discussions:- Integral test- harmonic series

Compute the center of mass of the solid, 1) Compute the center of mass of t...

1) Compute the center of mass of the solid of unit density 1 bounded (in spherical coordinates) by p=1 and by φ is greater than or equal 0 and less than or equal pi/4

Determine randomly generated bit string, Assume E is the event that a rando...

Assume E is the event that a randomly generated bit string of length 4 starts with a 1 and F is the event that this bit string consists of an even number of 1's. Are E and F indepe

Statistics, How do I choose a distribution test for a sample size of 60? Pr...

How do I choose a distribution test for a sample size of 60? Probability of rolling a 4 on a six sided die.

Trig, without using a calculator how would you know is cos theta(20) is gre...

without using a calculator how would you know is cos theta(20) is greater than cos theta (35)

Describe real numbers, Q. Describe Real numbers? Ans. There are a ...

Q. Describe Real numbers? Ans. There are a few different ways to describe real numbers. Without going into any of the very technical definitions used by mathematicians, I'

Relation between hieght, volume=(1/3)(pi)(radius of base)2(height) curved ...

volume=(1/3)(pi)(radius of base)2(height) curved surface area=(pi)(r)(l), r is radius of base and l is length of straight line connecting apex of cone with point on edge of base

Estimate how long did michael practice- algebra, Suppose that the number of...

Suppose that the number of hours Katie spent practicing soccer is represented through x. Michael practiced 4 hours more than 2 times the number of hours that Katie practiced. How l

Compare and contrast african immigrants, Compare and contrast African immig...

Compare and contrast African immigrants with our immigrant groups? How are they different? What are the implications of these differences for their adjustment to the larger society

Derivatives, application of derivatives in engg.

application of derivatives in engg.

Ellipsoid - three dimensional spaces, Ellipsoid Now here is the genera...

Ellipsoid Now here is the general equation of an ellipsoid. X 2 / a 2 + y 2 /b 2 + z 2 /c 2 = 1 Here is a diagram of a typical ellipsoid. If a = b = c afterw

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