Questions tagged [series]
The series tag has no usage guidance.
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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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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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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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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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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 a lot of sequences and series every day in our regular life.
I already found some examples such as the house numbers when you ...
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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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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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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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How can we explain intuitively the convergence and divergence of these two series?
It is known that $\displaystyle\sum_1^{\infty} \frac{1}{n^{1.000001}}$ converges while $\displaystyle\sum_{n\text{ is a prime number}}\frac{1}{n}$ diverges. Though we can logically prove these results,...
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Generating function example
I'm about to introduce the Generating Function concept to a couple of kids. The plan is just to roughly follow Herbert Wilf's Generatingfunctionology's first 12 pages, until Fibonacci numbers and Ch 1....
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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 ...
user13544
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Explaining Arithmetic Progression
So first introduction to a kid to arithmetic progression is when they see this, ...
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Emphasizing the decreasing condition in the Integral Test or in the AST (in Calculus II): is it worth the time?
The title is basically the question. But I guess I should expand a little. For the background, I'm teaching at a large public university in the US. Out student body is mixed in terms of their ...
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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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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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Finite sum of infinite series
I have two issues related to finite sum of infinite series,
1) How you would to describe 2 when you talk about the infinite geometric series 1+ 1/2 + 1/4 + 1/8 + .....
2) How you would compare using ...
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