0
$\begingroup$

Most series convergence questions are of a type where the term is explicitly specified : (Lets call them type 1 questions)

e.g. $\sum_{n=1}^{\infty} {a_{n}}$ where $a_n=$ some function of n. e.g. $\frac{1}{n}$ , $\frac{1}{n^2},\cdots etc.$

In question Prove that $\sum\limits_{n=1}^{\infty}\left (1-\dfrac{a_{n}}{a_{n+1}}\right)$ converges. , $a_n$ was not explicitly specified, instead it was only constrained by $a_n>0$. (lets call this type of questions type 2 questions).

My question is are there proper names for what we called type 1 and type 2 questions? e.g. in differentiation there are implicit and explicit differentiation.

Could we continue in similar fashion (by analogy or other means) and have convergence questions of type 3, 4, etc?

Is there a name for this type of generalisation/abstraction? What subject in mathematics would deal generalisation/abstraction specifically irregardless of topic ( e.g. DE's, Series, ...)

1 Answers 1

0

In mathematics, there are two kinds of questions :

  • Pratical or computational

we generally have explicit expressions . we use formulas, identities , criterions ... to solve.

  • Theoritical

To answer the question, you also need to use Theorems, lemmas....

  • 0
    I think what you mean is Applied and Abstract mathematics, the taxonomy I have questioned could very well be in Applied or Abstract mathematics. I am questioning types of abstractions/specialisation of each type and weather those abstractions could be more generalised specifically in context of convergence. – 2017-01-01