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Category Archives: real analysis
Zeno’s Paradoxes
The ancient Greek philosopher Zeno of Elea is known for proposing several paradoxes related to time, space, motion, and infinity. In this post we’ll focus on one of Zeno’s paradoxes and discuss how some ideas associated with calculus might or … Continue reading
Posted in calculus, infinity, logic, paradoxes, real analysis, sequences and series
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A Sequence of RightHand Riemann Sums Whose Limit Does Not Exist
Recently in my advanced calculus class one of my students asked a (perhaps the) key question when we hit the integration material: Why can’t you just use the limit of the righthand Riemann sums formed from equallyspaced partitions to define … Continue reading
Posted in calculus, real analysis
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Proof of the Recursive Identity for the Bernoulli Numbers
The Bernoulli numbers satisfy the recursive relationship . (Here, is the Iverson bracket, where evaluates to 1 if P is true and 0 if P is false.) The relationship can be used to calculate the Bernoulli numbers fairly easily. This post gives a … Continue reading
Posted in Bernoulli numbers, real analysis, recurrence relations
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Does Pointwise Convergence to a Continuous Function Imply Uniform Convergence?
Recently in my advanced calculus class we discussed how the uniform limit of a sequence of continuous functions is itself continuous. One of my students turned the question around: If the limit of a pointwise sequence of continuous functions is … Continue reading
Posted in real analysis
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Proof of the Irrationality of e
In a previous post I proved that is irrational. In this post I prove the irrationality of e. A proof of the irrationality of e must start by defining e. There are some different ways to do that. We’ll take e to be the unique … Continue reading
Posted in irrational numbers, real analysis
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Proving the Existence of Irrational Numbers
The ancient Greeks first proved the existence of irrational numbers by proving that is irrational. The proof is, as modern proofs of irrationality go, fairly simple. It is often the first example of a proof of irrationality that students see … Continue reading
Posted in irrational numbers, real analysis
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Euler Sums, Part II: A Symmetry Formula
A post from a few months ago gave a proof that In today’s post I’d like to prove a general symmetry formula for Euler sums like this one. Define and (Notice the upper index on the second sum is , … Continue reading
Posted in euler sums, real analysis, sequences and series
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Euler Sums, Part I
Sums of the form , where (i.e., the nth harmonic number of order p) are sometimes called Euler sums. There are a large number of interesting ways to evaluate these Euler sums, such as converting them to integrals, applying complex analysis, … Continue reading
Posted in calculus, euler sums, real analysis, sequences and series
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Which is bigger: π^e or e^π?
The question of whether or is larger is a classic one. Of course, with a calculator it is easy to see what the answer is. But how would you answer the question without a calculator? There are lots of interesting … Continue reading
Posted in real analysis
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