# Euler’s Series: problem worth trying for RMO or INMO or Madhava Mathematics Competition

Mengoli posed the following series to be evaluated:

$1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}}+\ldots$

Some great mathematicians, including Leibnitz, John Bernoulli and D’Alembert, failed to compute this infinite series. Euler established himself as the best mathematician of Europe (in fact, one of the greatest mathematicians in history) by evaluating this series initially, by a not-so-rigorous method. Later on, he gave alternative and more rigorous ways of getting the same result.

Question:

Can you show that the series converges and gets an upper limit? Then, try to evaluate the series.

Solution will be posted soon.

Cheers,

Nalin Pithwa.

# Inequalities and mathematical induction: RMO Homi Bhabha sample questions:

Prove by mathematical induction, the following:

1) $2^{n}(n!)^{2} \leq (2n)!$ for all $n \geq 1$.

2) Establish the Bernoulli inequality: If $(1+a)>0$, then $(1+a)^{n} \leq 1+na$ for all natural numbers greater than or equal to 1.

3) For all $n \geq 1$ with $n \in N$ prove the following by mathematical induction:

a) $\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \ldots + \frac{1}{n^{2}} \leq 2-\frac{1}{n}$

Solutions will be put up tomorrow!

Nalin Pithwa.

# Welcome to the Cambridge Mathematical Tripos

Introduction.

This is the first of what I hope will be a long series of posts aimed at providing back-up to first-year Cambridge mathematicians. This may seem a strange thing to do, since the Cambridge system of supervisions (classes taught on a one-to-two basis, usually discussing questions set by lecturers) already provides an excellent back-up to lectures. Do Cambridge undergraduates, who already have closer attention than in any other university I know about, really need even more help?

Well, perhaps they are lucky enough to need it less than mathematicians anywhere else, but there are several facts that convince me that even more can be done than is done already. Let me list a few of them.

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# Some “good” mathematics institutes in Mumbai/India

Just a quick note:

Now that there is a lot of awareness about Artificial Intelligence (AI), Machine Intelligence, Deep  Learning, Computer Vision, Data Science/Analytics etc., I feel that the “market value” of a pure mathematics degree or applied mathematics degree has risen very high in India also.

The following is just a personal view, which I would like to share with my readers/students (and their parents). (I  am not officially connected with any of the institutes mentioned below).

There are quite a few talented, hard-working, highly motivated students who  want to pursue mathematics at an undergraduate level in Mumbai (or elsewhere in India.) Some excellent options are as follows:

1. Mumbai; Pursue B.Sc. Maths at St. Xavier’s, Churchgate. And, then continue further  PG study at IIT’s, or TIFR, Mumbai or TIFR, Bangalore, or CMI, or ISI, or IISc, Bangalore.
2. Pune: S.P.College. B.Sc. Maths along with Madhava Competition. And, continue PG studies elsewhere.
3. Also, there are (the famous) IISER’s (Indian Institutes of Science Education and Research) offering 4 year BS in Mathematics and 5 year MS in Mathematics at six campuses in India. The admissions are based on IITJEE Mains, IITJEE Advanced or KVPY.
4. B.Sc. Mathematics programmes at : IISc, Bangalore; Chennai Mathematical Institute, (CMI), Chennai. The admission to IISc is through KVPY and admission to CMI is through its own entrance exam. CMI may be giving weightage to RMO  and INMO (of Homibhabha and TIFR).
5. Also, on the side, there is Indian Statistical Institute (ISI) offering B.Sc. degree in Statistics via its own entrance exam. The past set of questions papers are available for sale in a book form.

Thanks,

Nalin Pithwa