Zeno of Elea was a Greek philosopher who's most famous for the paradoxes that have been attributed to his name. While Zeno proposed his paradoxes to support, or discredit, various philosophical viewpoints, the paradox is frequently “solved” with a little bit of analysis.
There are many other paradoxes, some with more mathematical involvement than others, that can be fun to consider. It is also a fun exercise to try to create new ones, or modify those from Zeno to new situations.
Now I have a clear idea that the summation will be:
Which is a geometric series that converges.
Taking that quantity and dividing by the original deposit is a quantity called the credit multiplier. Many different fields in the study of economics look at multipliers as a method of analysis or comparison.
Series Without Negative Terms
Which is divergent.
Series With Odd or Even Negative Terms
Will not be written the same way as the series:
Therefore, some tricky work must be done with the indexing. Often times there will be numbers added or subtracted to the indexing of the summation, the exponent or anywhere else to get signs and numbers to agree. In the case of the above series, each are an alternating harmonic series, so we know we will start out with:
In the first case, the first term is positive, so we need the exponent to be even for the first term. Therefore we need to add one. No such addition is needed for the second series, as the negatives work with the regular indexing.
It is a good challenge for students to try to think up series that skip terms, alternate signs and other tricks that may require a bit of puzzle solving to write out.
Ratio Test, Root Test and Summary of Tests
An added challenge for students can using some of the techniques of calculus to not only determine convergence, but find the sum of the series.
Now finding the value of the sum is a little bit tricky. This is a nice application of the method of partial fractions outside of integrals, as wel will need to split up that denominator to find a solution.]
Substituting in and then splitting up the summation:
Now we can change the index of each to eliminate the terms in the denominator.
Finding ways to approximate functions with power series is a tough task for students. Here is some additional reinforcement with another standard problem.
There are two tricks here. First of all we want to try to convert to a series at some point and usually the easiest way is to use a geometric series. Also, a common trick to get logarithms into the form of a geometric series is to use the derivative. This gives a fraction that can be manipulated into the correct form:
Integrate both sides:
Checking for the radius of convergence:
So taking the limit:
Taylor and MacLaurin Series
We can examine why this was important with the following questions. Remember, the whole advantage of Taylor series is that it allows nearly any function to be calculated as a polynomial. This has two implications; first, this is how computers and calculators compute transcendental functions. Second, if you do not have a caculator, or you are attempting to find a value that is previously unknown so it does not appear in a table, the first number of terms in a taylor sum will allow you to find that value.
A student may ask how we knew to take the binomial expansion of that particular function. There is no really good answer, as all the time mathematicians are asserting that something is true, and then proving it later, seemingly picking ideas out of thin air. In fact, we will make a doozy of an assumption later. Sometimes guess and check can tell us where we need to go. Here, we are taking a function and that is close to some form of our original function.
But we still need to show that Machin’s formula is correct. We will start by making the assertion that:
To show that this is the case, use the angle sum formula for tangent:
Much the same way, we now need to show:
This is easiest to show in two steps. First show:
It should all come together now substituting back to the top. It is also useful to remember that negatives inside of a tangent become negatives outside due to symmetry. It is common for students to believe that taylor series are antiquated, made obsolete by the calculator. As it actually stands, someone has to program all of those functions into the calculator, and the most common technique is to use the equivalent taylor series. Our calculators would not know how to take the tangent of an angle otherwise. This is an elegant way to compute many digits of pi without extreme computer power.