Questions tagged [series]

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Motivating example for sequences, sums and limits in high school

I currently work as a substitute teacher at a local high school and the topic in one of the classes is sequences, series and limits. Because I always disliked learning about a topic without having ...
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1 vote
1 answer

Intuition explanation about Lebesgue measure zero of the rational numbers [closed]

This is a question about the intuition of the rational number having measure zero. Let us consider followng proof: Let $I = [0,1]$ and $Q = \mathbb Q \cap I$ and let $\lambda$ be the Lebesgue measure. ...
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4 votes
5 answers

What strategy for picking convergence tests for series do you teach?

Without getting bogged down in details, I'll list the names only. It seems that the strategy I generally use is this: Divergence test first Is it a recognizable form? p-series or geometric? a) No ...
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2 votes
4 answers

Proof that convergent Taylor Series converge to actual value of function

Taylor series (or Maclaurin Series) are the only way to get values for some functions, such as $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{t^2} dt = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}...
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  • 121
5 votes
8 answers

Comparison Tests in Calculus

How should I teach Comparison Tests in Calculus II, and why? Note that I will cover comparison tests in some way, and students will be expected to justify their answers to questions about series ...
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9 votes
7 answers

Examples of arithmetic and geometric sequences and series in daily life

In this part of the course I am just trying to show that we actually see alot of sequences and series everyday in our daily life. I already found some examples such as the housenumbers when you drive ...
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8 votes
5 answers

Geometric Series Formula and Calculus

Is there any good reason that in educational materials, I consistently see the formula for calculating geometric series in canonical form as: $$\sum_{k=0}^{n-1} ar^k = a \frac{1-r^n}{1-r}$$ While an ...
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  • 2,588
11 votes
6 answers

How can I convince students that Fourier series are useful?

Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are ...
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13 votes
8 answers

What are some fun/nonstandard examples of arithmetic/geometric series?

I am teaching those topics (arithmetic/geometric series) just now, and want some not so standard (fun) examples, which can be used essentially at high school/beginning calculus level. I'm ...
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29 votes
11 answers

For calculus students, what should be the intuition or motivation behind series?

I've noticed that series are one of the most difficult portions of calculus for new students to learn. I think the level of abstraction has to do with this. Limits, derivatives, and integrals, as ...
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32 votes
11 answers

How can I teach my students the difference between a sequence and a series?

Sequences and series are related concepts but differ extremely from one another. I feel that students in integral calculus frequently mix them up. Part of the problem is that: Sequences are usually ...
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