understanding limit and continuity - brief sketch of a Cheenta Session

The limit and continuity is considered as the most important 'idea' in Analysis (for our purpose 'calculus'). Unfortunately deep understanding of these to concepts are beyond the reach of school level mathematical culture of India. Cheenta presents a series of intensive classes that puts the knowledge thirsty student in the right track.

1. Understanding monotonic and bounded sequences by definition and examples.
2. Considering the optional existence of analytical expression (eg. series of primes) and graphical expression (Dirichlet Sequence).
3. Considering the presence of monotonicity and boundedness in the same sequence (Xeno's paradox, eye of Horas). Considering the case of \((1+ \frac{1}{n})^n\) and  \((1+ \frac{1}{n})^{n+1}\)
4. The epsilon definition of limit of a sequence
1. The number a is said to be the limit of a sequence ( \(y_n\) ) if for any positive number \(\epsilon\) there is a real number N such that for all n > N the following inequality holds: | \(y_n\) - a | < \(\epsilon\).
5. Some problems concerning trivial limit calculations.
6. Some theorems related to limit of a sequence:
1. If a sequence has a limit it is bounded.
2. If a sequence is both bounded and monotonic, it has a limit. (Weierstrass)
3. A convergent sequence has only one limit.
7. A schematic approach to handle boundedness, monotonicity, convergency.
8. Some more theorems on limit of a sequence:
1. If a sequence (\(y_n\) ) and (\(z_n\) are convergent (we denote their limits by a and b respectively), a sequence (\(y_n + z_n\) is convergent too, it's limit being a+b.
2. Converse of the above theorem is not necessarily true. Very important point and is illustrated by an example. (if sum is convergent, then both are convergent or both are divergent but not otherwise).
3. \(\displaystyle\lim_{n\to\infty}c y_n = c \displaystyle\lim_{n\to\infty}y_n\)
9. Understanding infinitesimal sequences (definition)
10. Some theorems concerning infinitesimals
1. If (\(y_n\)) is a bounded sequence and (\(\alpha_n\) is infinitesimal, then (\(y_n \alpha_n\) is infinitesimal as well.
2. A sequence (\(y_n z_n\)) is convergent to ab if the sequences (\(y_n\)) and (\(z_n\)) is convergent to a and b respectively.
3. If a sequence (\(y_n\) ) and (\(z_n\) are convergent (we denote their limits by a and b respectively when b is not 0), a sequence (\(\frac{y_n}{z_n}\) is convergent too, it's limit being \(frac{a}{b}\).
11. Understand elimination, or addition, and any other change of a finite number of terms of a sequence do not affect either it's convergence or it's limit (if the sequence is convergent). We consider the case for the change of infinite terms as well.
12. Finally we take up problem solving from I.A. Maron (testing convergence of sequences).

This discussion concludes the first session of Limit and continuity concept building.