APPLICATIONS OF SERIES
CALCULUS II FALL 2013 SECTION 3 SITARAMAN
Sequences and series appear throughout Nature. Starting with the naturl numbers (which is itself a sequence), sequences such as arithmetic sequences, geometric sequences, fibonacci sequences appear everywhere in life. Any time you need to add numbers that appear in a sequence, series make an appearance.
Below you will see some examples and applications. Click on each hyperlinked item to see more information about it. I won't talk about arithmetic sequences or Fibonacci sequences here. These are very basic and popular and you can find a lot of material on the web about these. Besides, if you add up the terms of Fibonacci or arithmetic sequences, the resulting infinite series do not add up to a finite sum.
GEOMETRIC SERIES
After arithmetic series, geometric series are one of the most prominent in life. Any time you invest money or take out a loan, and you want to add up the periodic payments or returns over several years, geometric series are involved.
Derivation of mortgage monthly payment formula from my Spring 2012 college algebra 2 class
Geometric series and investment
From UC Riverside Prof. Bergner's website:
How much money
do you need to have saved for retirement so that you can withdraw a fixed
amount of money each year for 30 years? If the lottery promises to pay you
10 million dollars over 10 years, how much is that worth today?
Geometric Series and Annuities
Geometric series and present value of a stock
How are stock prices determined? It is based on calculating how much the value of a stock will grow over time if it grows in a certain rate based on how much dividend the company will pay out, how the company will perform, etc.,
Here is a calculation of this from Sloppy Joe capital:
Current price of stock
Notice how they have used the formula (they even prove it along the way) for the first n terms of a geometric series.
p-Series
When p=1 we have what is called the harmonic series.
This is the sum of the reciprocals of 1,2,3,...
As mentioned in class, the origin of the word "harmonic" comes from music, where it was noted that the various notes have frequencies related using ratios of natural numbers.
Here is a discussion on harmonic series and music from Earlham College:
Harmonic Series and Music
When p is bigger than 1 we get convergent series.
The values of these series come up in many places in higher math and physics.
In fact, the function obtained by thinking of the p-series as a function of p
is called the Riemann Zeta function, where p can be any complex number.
Here area couple of discussions on Zeta functions in Physics:
A rather advanced discussion on zeta function and physics from University of Illinois, Urbana-Champaign
A more general discussion, in the form of slide-show, from my alma mater CalTech:
Numbers and Riemann zeta function in Physics