Homework 11, Problem 8

Professor Covault,

For problem 8, we are asked to find the general series of N bricks representing the total distance extended beyond the leading edge of the bottom brick. I found it to be the sum of L/(2i) were i starts at 1 and N=N. For part B, we are asked to evaluate the series with N=1000. I've learned some of the rules of sigma notation from my current calc. class, but I can't seem to find a general formula for when i is in the denominator. My only guess would be to go through and manually add them all up. But that would take a very long time. What would be the next step? Is there a shortcut I don't know/I'm forgetting?

A Long time indeed.

There are several clever ways to do this but if you want to simply
use a "fast calculator" check out this link:

[www.math.utah.edu]

-CC

I'm running into a few roadblocks:

1- for part C, where we are to find the distance when N=1 million and when N=1 trillion, I am able to find when N=1 million using the website provided, but it will not provide me an answer for when N=1 trillion. I'm not sure whether this could be due to an overload by the program (which is likely) or by the fact that N has already reached it's limit.

2- part d asks us for the limit of the series. I was going to try this for part c, but I'm not sure how to go about doing this. Normally for a general series, I would use the formula for that particular case and take its limit as N approaches infinity.

The reason the calculator breaks is because when you ask for the
function for N= a trillion it has to add a trillion numbers.

This is a very interesting series. Basically for large N
the series approaches a value which is the \ln(N) plus a
fixed small constant = about 0.577 calle the Euler-Mascheroni constant.
See this link So for large N the value of the function is 0.577 plus ln(N).
This works to very high precision for any value of N that is much above
100.

Note that since the log function diverges at infinity so does this
function.

For more of any and everything you ever did not really want to know
about this series check out this link.

What this tells you is that the series does diverge but it diverges very slowly.

-CC

Is this the concept that is applied to an arch bridge? Like if the height of your bricks were nearly zero, wouldn't it make the shape of an arch and hold true to this concept?

Umm, sort of. For an arch then you have one extra benefit in that
there is a normal force on horizontally matching bricks that gives a
torque that counters the torque due to gravity that wants to topple
the bricks over. So its easier to build an arch than it is to build
something that hangs by itself...

-CC