Famous harmonic series questions

Question 1:

The thirteenth century French polymath Nicolas Oresme proved that the harmonic series 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \ldots does not converge. Prove this result.

Question 2:

Prove that \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \ldots does not converge.

Question 3:

Prove that 1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \ldots does not converge.

Even if you solve these all on your own, you won’t achieve the glory of that French polymath, but you will have “re-discovered” some “elements of truth of analysis” …You can give yourself a pat on the back!

Cheers,

Nalin Pithwa.

Infinity of primes — proof using calculus

Let us look at a proof that the number of primes is infinite — a proof that uses only elementary calculus.

Let \pi (x) = \# \{ p \leq x: p \in P\} be the number of primes that are less than or equal to the real number x. (Note: here P is the set of primes). We number the primes P = \{ p_{1}, p_{2}, p_{3, \ldots}\} in increasing order. Consider the natural logarithm \log {x} defined as \log {x}=\int_{1}^{x}\frac{1}{t}dt.

Now we compare the area below the graph of f(t)=\frac{1}{t} with an upper step function. Thus, for n \leq x < n+1 we have

\log {x} \leq 1+\frac{1}{2}+\frac{1}{3}+ \ldots + \frac{1}{n-1}+\frac{1}{n} \leq \Sigma \frac{1}{m} where the sum extends over all m \in N which have only prime divisors p \leq x.

Since every such m can be written in a unique way as a product of the form \Pi_{p \leq x}p^{k_{p}}, we see that the last sum is equal to

\Pi_{{p \in P}_{p \leq x}}(\Sigma_{k \geq 0}\frac{1}{p^{k}})

The inner sum is a geometric series with ratio \frac{1}{p}, hence,

\log {x} \leq \Pi_{{p \in P}_{p \leq x}}\frac{1}{1-\frac{1}{p}}=\Pi_{{p \in P}_{p \leq x}}\frac{p}{p-1}=\Pi_{k=1}^{\pi(x)}\frac{p_{k}}{p_{k}-1}

Now, clearly p_{k} \geq k+1 and thus

\frac{p_{k}}{p_{k}-1}=1+ \frac{1}{p_{k}-1} \leq 1 + \frac{1}{k}=\frac{k+1}{k}

and therefore,

\log {x}\leq \Pi_{k=1}^{\pi(x)}\frac{k+1}{k}=\pi(x)+1.

Everybody knows that \log {x} is not bounded, so we conclude that \pi(x) is unbounded as well, and so there are infinitely many primes. QED.

Ref: Proofs from THE BOOK (Martin Aigner and Gunter M. Ziegler) (Third Edition).

More about primes later,

Nalin Pithwa

 

 

 

 

Real Numbers, Sequences and Series: part 9

Definition.

We call a sequence (a_{n})_{n=1}^{\infty} a Cauchy sequence if for all \varepsilon >0 there exists an n_{0} such that |a_{m}-a_{n}|<\varepsilon for all m, n > n_{0}.

Theorem:

Every Cauchy sequence is a bounded sequence and is convergent.

Proof.

By definition, for all \varepsilon >0 there is an n_{0} such that

|a_{m}-a_{n}|<\varepsilon for all m, n>n_{0}.

So, in particular, |a_{n_{0}}-a_{n}|<\varepsilon for all n > n_{0}, that is,

a_{n_{0}+1}-\varepsilon<a_{n}<a_{n_{0}+1}+\varepsilon for all n>n_{0}.

Let M=\max \{ a_{1}, \ldots, a_{n_{0}}, a_{n_{0}+1}+\varepsilon\} and m=\min \{ a_{1}, \ldots, a_{n_{0}+1}-\varepsilon\}.

It is clear that m \leq a_{n} \leq M, for all n \geq 1.

We now prove that such a sequence is convergent. Let \overline {\lim} a_{n}=L and \underline{\lim}a_{n}=l. Since any Cauchy sequence is bounded,

-\infty < l \leq L < \infty.

But since (a_{n})_{n=1}^{\infty} is Cauchy, for every \varepsilon >0 there is an n_{0}=n_{0}(\varepsilon) such that

a_{n_{0}+1}-\varepsilon<a_{n}<a_{n_{0}+1}+\varepsilon for all n>n_{0}.

which implies that a_{n_{0}+1}-\varepsilon \leq \underline{\lim}a_{n} =l \leq \overline{\lim}a_{n}=L \leq a_{n_{0}+1}+\varepsilon. Thus, L-l \leq 2\varepsilon for all \varepsilon>0. This is possible only if L=l.

QED.

Thus, we have established that the Cauchy criterion is both a necessary and sufficient criterion of convergence of a sequence. We state a few more results without proofs (exercises).

Theorem:

For sequences (a_{n})_{n=1}^{\infty} and (b_{n})_{n=1}^{\infty}.

(i) If l \leq a_{n} \leq b_{n} and \lim_{n \rightarrow \infty}b_{n}=l, then (a_{n})_{n=1}^{\infty} too is convergent and \lim_{n \rightarrow \infty}a_{n}=l.

(ii) If a_{n} \leq b_{n}, then \overline{\lim}a_{n} \leq \overline{\lim}b_{n}, \underline{\lim}a_{n} \leq \underline{\lim}b_{n}.

(iii) \underline{\lim}(a_{n}+b_{n}) \geq \underline{\lim}a_{n}+\underline{\lim}b_{n}

(iv) \overline{\lim}(a_{n}+b_{n}) \leq \overline{\lim}{a_{n}}+ \overline{\lim}{b_{n}}

(v) If (a_{n})_{n=1}^{\infty} and (b_{n})_{n=1}^{\infty} are both convergent, then (a_{n}+b_{n})_{n=1}^{\infty}, (a_{n}-b_{n})_{n=1}^{\infty}, and (a_{n}b_{n})_{n=1}^{\infty} are convergent and we have \lim(a_{n} \pm b_{n})=\lim{(a_{n} \pm b_{n})}=\lim{a_{n}} \pm \lim{b_{n}}, and \lim{a_{n}b_{n}}=\lim {a_{n}}\leq \lim {b_{n}}.

(vi) If (a_{n})_{n=1}^{\infty}, (b_{n})_{n=1}^{\infty} are convergent and \lim_{n \rightarrow \infty}b_{n}=l \neq 0, then (\frac{a_{n}}{b_{n}})_{n=1}^{\infty} is convergent and \lim_{n \rightarrow \frac{a_{n}}{b_{n}}}= \frac{\lim {a_{n}}}{\lim{b_{n}}}.

Reference: Understanding Mathematics by Sinha, Karandikar et al. I have used this reference for all the previous articles on series and sequences.

More later,

Nalin Pithwa

 

Real Numbers, Sequences and Series: part 8

Definition. A sequence is a function f:N \rightarrow \Re. It is usual to represent the sequence f as (a_{n})_{n=1}^{\infty} where f(n)=a_{n}.

Definition. A sequence (a_{n})_{n=1}^{\infty} is said to converge to a if for each \varepsilon>0 there is an n_{0} such that

|a_{n}-a|<\varepsilon for all n > n_{0}.

It can be shown that this number is unique and we write \lim_{n \rightarrow \infty}a_{n}=a.

Examples.

(a) The sequence \{ 1, 1/2, 1/3, \ldots, 1/n \} converges to 0. For \varepsilon>0, let n_{0}=[\frac{1}{\varepsilon}]+1. This gives

|\frac{1}{n}-0|=\frac{1}{n}<\varepsilon for all n>n_{0}.

(b) The sequence \{ 1, 3/2, 7/4, 15/8, 31/16, \ldots\} converges to 2. The nth term of this sequence is \frac{2^{n}-1}{2^{n-1}}=2-\frac{1}{2^{n-1}}. So |2-(2-\frac{1}{2^{n-1}})|=\frac{1}{2^{n-1}}. But, 2^{n-1} \geq n for all n \geq 1. Thus, for a given \varepsilon >0, the choice of n_{0} given in (a) above will do.

(c) The sequence (a_{n})_{n=1}^{\infty} defined by a_{n}=n^{\frac{1}{n}} converges to 1.

Let n^{\frac{1}{n}}=1+\delta_{n} so that for n >1, \delta_{n}>0. Now, n=(1+\delta_{n})^{n}=1+n\delta_{n}+\frac{n(n-1)}{2}(\delta_{n})^{2}+\ldots \geq 1+\frac{n(n-1)}{2}(\delta_{n})^{2}, thus, for n-1>0, we have \delta_{n} \leq \sqrt{\frac{2}{n}}. For any \varepsilon > 0, we can find n_{0} in N such that n_{0}\frac{(\varepsilon)^{2}}{2}>1. Thus, for any n > n_{0}, we have 0 \leq \delta_{n} \leq \sqrt{\frac{2}{n}} \leq \sqrt{\frac{2}{n_{0}}}<\varepsilon. This is the same as writing

|n^{\frac{1}{n}}-1|<\varepsilon for n > n_{0} or equivalently, \lim_{n \rightarrow \infty}n^{\frac{1}{n}}=1

(d) Let a_{n}=\frac{2^{n}}{n!}. The sequence (a_{n})_{n=1}^{\infty} converges to o. Note that

a_{n}=\frac{2}{1}\frac{2}{2}\frac{2}{3}\ldots\frac{2}{n}<\frac{4}{n} for n>3.

For \varepsilon>0, choose n_{1} such that n_{1}\varepsilon>4. Now, let n_{0}=max \{ 3, n\} so that |a_{n}|<\varepsilon for all n > n_{0}.

(e) For a_{n}=\frac{n!}{n^{n}}, the sequence (a_{n})_{n=1}^{\infty} converges to 0. (Exercise!)

In all the above examples, we somehow guessed in advance what a sequence converges to. But, suppose we are not able to do that and one asks whether it is possible to decide if the sequence converges to some real number. This can be helped by the following analogy: Suppose there are many people coming to Delhi to attend a conference. They might be taking different routes. But, as soon as they come closer and closer to Delhi the distance between the participants is getting smaller.

This can be paraphrased in mathematical language as:

Theorem. If (a_{n})_{n=1}^{\infty} is a convergent sequence, then for every \varepsilon>0, we can find an n_{0} such that

|a_{n}-a_{m}|<\varepsilon for all m, n > n_{0}.

Proof. Suppose (a_{n})_{n=1}^{\infty} converges to a. Then, for every \varepsilon we can find an n_{0} such that

|a_{n}-a|<\varepsilon/2 for all n > n_{0}.

So, for m, n > n_{0}, we have

|a_{m}-a_{n}|=|a_{n}-a+a-a_{m}|\leq |a_{n}-a|+|a-a_{m}|<\varepsilon. QED.

The proof is rather simple. This very useful idea was conceived by Cauchy and the above theorem is called Cauchy criterion for convergence. What we have proved above tells us that the criterion is a necessary condition for convergence. Is it also sufficient? That is, given a sequence (a_{n})_{n=1}^{\infty} which satisfies the Cauchy criterion, can we assert that there is a real number to which it converges? The answer is yes!

We call a sequence (a_{n})_{n=1}^{\infty} monotonically non-decreasing if a_{n+1} \geq a_{n} for all n.

Theorem. A monotonically non-decreasing sequence which is bounded above converges.

Proof. Suppose (a_{n})_{n=1}^{\infty} is monotonic and non-decreasing. We have a_{1} \leq a_{2} \leq \ldots \leq a_{n} \leq a_{n+1} \leq \ldots. Since the sequence is bounded above, \{ a_{k}: k=1, \ldots\} has a least upper bound. Let it be a. By definition, a_{n} \leq a for all n, but for \varepsilon>0 there is at least one n_{0} such that a_{n_{0}}+\varepsilon>a. Therefore, for n > n_{0}, a_{n}+\varepsilon \geq a_{n_{0}}+\varepsilon>a\geq a_{n}. This gives us |a_{n}-a|<\varepsilon for all n \geq n_{0}. QED.

We can similarly prove that: A monotonically non-increasing sequence which is bounded below is convergent.

Suppose we did not have the condition of boundedness below or above for a monotonically non-increasing or non-decreasing sequence respectively, then what would happen? If a sequence is monotonically non-decreasing and is not bounded above, then given any real number M>0 there exists at least one n_{0} such that a_{n_{0}}>M, and hence a_{n}>M for all n > n_{0}. In such a case, we say that (a_{n})_{n=1}^{\infty} diverges to \infty. We write \lim_{n \rightarrow \infty}a_{n}=\infty. More generally, (that is, even when the sequence is not monotone), the same criterion above allows us to say that \{ a_{n}\}_{n=1}^{\infty} diverges to \infty and we write \lim_{n \rightarrow \infty}a_{n}=\infty. We can similarly define divergence to -\infty.

We make a digression here: In case a set is bounded above, then we have the concept of least upper bound. For any set A or real numbers, we define supremum of A as

sup \{ x: x \in A\}=sup A = least \hspace{0.1in} upper \hspace{0.1in}bound \hspace{0.1in} of \hspace{0.1in} A, if A is bounded above and is equal to \infty if A is not bounded above.

Similarly, we define infimum of a set A as

inf \{ x: x \in A\}= inf A= greatest \hspace{0.1in} lower \hspace{0.1in} bound \hspace{0.1in} of \hspace{0.1in}A, if A is bounded below, and is equal to -\infty, if A is not bounded below.

This would help us to look for other criteria for convergence. For any bounded sequence (a_{n})_{n=1}^{\infty} of real numbers, let

b_{n} = \sup \{ a_{n}, a_{n+1}, a_{n+2}, \ldots\}=\sup \{ a_{k}: k \geq n\}.

It is clear that (b_{n})_{n=1}^{\infty} is now a non-increasing sequence. So, it converges. We set

\lim_{n \rightarrow \infty}b_{n}=limit \hspace{0.1in} superior \hspace{0.1in} of \hspace{0.1in} the \hspace{0.1in} sequence \hspace{0.1in} (a_{n})_{n=1}^{\infty}= \lim \sup a_{n}= \overline{\lim}_{n}a_{n},

Similarly, if we write

a_{n}=\inf \{ a_{n}, a_{n+1}, \ldots\}=\inf \{ a_{k}: k \geq n\},

then (a_{n})_{n=1}^{\infty} is a monotonically non-decreasing sequence. So we write

\lim_{n \rightarrow \infty}c_{n}=limit \hspace{0.1in} inferior \hspace{0.1in} of \hspace{0.1in} the \hspace{0.1in} sequence \hspace{0.1in} (a_{n})_{n=1}^{\infty}= \lim \inf a_{n}= \underline{\lim}_{n}a_{n},

We may not know a sequence to be convergent or divergent, yet we can find its limit superior and limit inferior.

We have in fact the following result:

Theorem. 

For a sequence (a_{n})_{n=1}^{\infty} of real numbers, \underline{\lim}a_{n} \leq \overline{\lim}a_{n}.

Further, if l=\underline{\lim}a_{n}, and L=\overline{\lim}a_{n} are finite, then l=L if and only if the sequence is convergent.

Proof.

It is easy to see that l \leq L. Now, suppose l, L are finite.If l=L are finite. If l=L, then for all \varepsilon >0 there exists n_{1}, n_{2} such that

l - \varepsilon< \sup_{k \geq n} a_{k}< l + \varepsilon for all n \geq n_{1} and l - \varepsilon < \inf_{k \geq n} < l + \varepsilon for all n > n_{2}.

Thus, with n_{0}=max{n_{1},n_{2}} we have

l - \varepsilon<a_{n}< l + \varepsilon for all n > n_{0}.

This proves that equality holds only when it is convergent. Conversely, suppose (a_{n})_{n=1}^{\infty} converges to a. For every \varepsilon > 0, we have n_{0} such that a-\varepsilon<a_{n}<a+\varepsilon for all n > n_{0}. Therefore, \sup_{k \geq n}a_{k} \leq a+\varepsilon and

a-\varepsilon \leq \inf a_{n} for all n \geq n_{0}. Hence, a-\varepsilon \leq < L \leq a+\varepsilon. Since this is true for every \varepsilon > 0, we have L=l=a.

QED.

Real Numbers, Sequences and Series: Part 6

Theorem:

Given x \in \Re_{+} and n \in N, we can find y \in \Re such that x=y^{n}.

Proof:

Let A={u \in \Re_{+}: u^{n}<x}. If x<1, then x^{n} <x and hence x \in A. On the other hand, if x \geq 1, then 1/2 \in A. Thus, A is non-empty. Next, observe that if v^{n}>x, then v is an upper bound of A. In particular, \{ 1+x\} is an upper bound of A.

By the least upper bound property, A admits a least upper bound. Let us denote it by y. We will rule out the possibilities y^{n}>x and y^{n}<x implying that x=y^{n}.

If y^{n}<x, let a=\frac{x-y^{n}}{nx} and z=y(1+a). It can be checked that z^{n}<x so that z \in A. But, y <x, contradicting the fact that y is the least upper bound of A. (we have used the inequalities 7 and 8 in the previous blog).

On the other hand, if y^{n}>x, let \frac{y^{n}-x}{ny^{n}} and w=y(1-b). Again, it can be verified that w^{n}>x and hence w is an upper bound of A. But, w<y, contradicting the fact that y is the least upper bound of A. Hence, y^{n}=xQED.

In particular, we see that there is an element \alpha \in \Re such that \alpha^{2}=2 and hence also (-\alpha)^{2}=2 which means that the equation x^{2}=2 has two solutions. The positive one of those two solutions is \sqrt{2}. In fact, the above theorem has guaranteed its extraction of the square root, cube root, even nth root of any positive number. You could ask at this stage, if this guarantees us extraction of square root of a negative number. The answer is clearly no. Indeed, we have

x^{2} \geq 0 for x \in \Re.

Remark.

We can further extend \Re to include numbers whose squares are negative, which you know leads to the idea of complex numbers.

We have shown that Q is a subset of \Re. One can also show that between any two distinct real numbers there is a rational number. This naturally leads to the decimal representation of real number: Given any real number x and any q \in N, we can get a unique a_{0} \in Z and unique a_{1},a_{2}, \ldots a_{q} \in N such that 0 \leq a_{1} \leq a_{2} \ldots a_{q} \leq 9 and

|x-(a_{0}+a_{1}/10+a_{2}/100+\ldots + \frac{a_{q}}{10^{q}})|<\frac{1}{10^{q}}

You are invited to try to  prove this familiar decimal representation.

If we have a terminating decimal representation of a real number, then surely it is rational.But, we know that rationals like 1/3, 1/7, 1/11, do not have a terminating decimal expansion.

It is clear that the decimal representation of \sqrt{2} cannot terminate as it is not rational. There are many elements of \Re which are not in Q. Any element of \Re which is not in is called an irrational number, and irrational numbers cannot have terminating decimal representation.

More later,

Nalin Pithwa

Real numbers, sequences and series: part VI

No study is complete without solving problems on your own. Below are the exercises related to part V.

Exercises.

Using the properties of real numbers, show that

1) \alpha <0 or \alpha=0 or \alpha>0.

2) If \alpha \neq 0, (-\alpha)^{2}=(\alpha)^{2}>0.

3) if 0<a\leq b, then \frac{1}{b} \leq \frac{1}{a}.

4) If 0 \leq a,b, then (1-a)(1-b) \geq 1-a-b.

5) If 0<a<1, then a^{n}<1 for any positive integer n.

6) For 0<a,b, a^{n}<b^{n} implies a<b.

7) If 0 \leq a, then (1+a)^{n}\geq 1+na, and (1-a)^{n} \geq 1-na, and

8) If 0<a<\frac{1}{n}, then (1+a)^{n}<\frac{1}{(1-a)^{n}}<\frac{1}{1-na}

9) Every set A \subseteq \Re that is bounded below admits a greatest lower bound.

More later,

Nalin Pithwa

Real numbers, sequences and series: part V

Real Numbers.

Our scheme of real numbers has so far been extended to rational numbers so that we could not only add and subtract without any hindrances, but could also multiply and divide without any restriction, save division by zero. But, when it comes to extraction of square roots, it seems “incomplete”. For example, there is no rational number whose square gives 2. This question crops up when we try to relate numbers to geometry.

We have seen before that we can always represent a rational number on a line. Now suppose we have a square with sides of unit length. By the theorem of Pythagoras, the square of the length of diagonal is equal to 1^{2}+1^{2}=2. We can also see that the length of the diagonal can be geometrically represented on the line. Now, we ask here, does there exist a rational number whose square is 2, or equivalently, does the equation x^{2}=2 have a solution in rational numbers? This amounts to asking if we can find two non-zero integers p and q such that

p^{2}=2q^{2}Equation I.

We can demonstrate that there are no such integers. Suppose that there were two such integers p and q satisfying equation I. We may assume without loss of generality, that p and q have no common factors. If there were any, we could cancel them from both the sides of equation I till there were no common factors. Now as there are no common factors between p and q, for equation I to hold, 2 must be a factor of p^{2}, and hence, of p. So we may write

p=2m Equation 2.

Substituting this in Equation 2, we get 2m^{2}=q^{2}Equation 3.

As before, we conclude that for equation 3 to hold, 2 must divide q giving q=2n for some integer n. This contradicts our assumption that p and q have no common factors. This only proves that equation I has no solution in integers (except p=0, q=0). Thus, the length of the diagonal of the unit square, though it has a point representing it on the line, does not correspond to a rational number. This shows that is not large enough to accommodate a number such as the length of the diagonal of a unit square. So we must extend Q. We could simply include the new number which we write as \sqrt{2} in the scheme along with Q. But, then this ad hoc extension may not stand up to all our demands to include newer and newer numbers which arise out of algebraic equations, e.g.,\sqrt[3]{6}, etc. Even if we somehow include these numbers in our scheme, we must know how to perform arithmetic operations like addition, subtraction, multiplication and division in it. Also, how does one decide which of a given pair of new numbers is smaller? There are many ways in extending to accommodate all these new numbers. We illustrate one of the simplest ways of doing this.

Let E=\{ a \in Q: a^{2}<2\}. It is clear that E has an upper bound, for example, 2. Next, we note that if y \in E is an upper bound of E, then there exists n \in N with n(2-y^{2})>1+2y (we can always choose an n because of the Archimedean property of Q). Then, one has 2-(y+\frac{1}{n})^{2}=(2-y^{2})-\frac{1}{n}(2y+\frac{1}{n}) \geq (2-y^{2})-\frac{2y+1}{n}>0, showing that y^{2}<(y+\frac{1}{n})^{2}<2 and thus no upper bound of E can be in E. Let x \in Q be one such upper bound of E, then so is \frac{x}{2}+\frac{1}{x} since (\frac{x}{2}+\frac{1}{x})^{2}-2=(\frac{x}{2}-\frac{1}{x})^{2}>0. On the other hand, x-(\frac{x}{2}+\frac{1}{x})=\frac{x^{2}-2}{2x}>0 since x \not\in E. Therefore, the E has no least upper bound in Q. 

This tells us what to do. We now extend the field of rationals to a larger field containing it which has all properties of along with an additional property, called the least upper bound property and abbreviated as lub property, namely, every set that is bounded above has a least upper bound. 

More precisely, we postulate that there is a field (\Re, +, .), \Re containing Q and satisfying all the properties of Q listed above and an additional one:

The least upper bound property (lub): If A \subseteq \Re is bounded above, then A admits a least upper bound, that is, (1) there is a \gamma \in \Re such that \alpha \leq \gamma for every \alpha \in A and (2) for every \gamma^{'}< \gamma, there is a \beta \in A such that \gamma^{'} < \beta.

Remark:

As we constructed out of Z, there are ways of constructing \Re out of Q. However, an explicit construction of \Re from is beyond the scope of the present blog. Nevertheless, we are going to use the listed properties of \Re in all subsequent discussions and deduce many interesting consequences.

Since \Re has all the properties of including the order property, we write (as for Q),

\Re_{+}=\{ \alpha \in \Re|\alpha >0\}

and this has the same properties as Q_{+}. Now, we use the same notation like \leq, <, \geq, > for elements of \Re_{+}, exactly as we did for the elements of Q. As before, we write \beta > \alpha or \beta \geq \alpha according as \alpha < \beta or \alpha \leq \beta.

Clearly, 1, 1+1, 1+1+1, \ldots belong to \Re and represent the natural numbers. So, one sets up a one to one correspondence between n \in N and 1+1+1+\ldots+1 (n times) to conclude that N \subset \Re. Writing -n as the additive inverse of n \in N, of course 0 \in \Re, we see that Z \subseteq \Re. Since \Re is a field, for m,n \in Z, n \neq 0, we must have

m.n^{-1} \in \Re

where n^{-1} is the multiplicative inverse of n. We agree to write m.n^{-1}=\frac{m}{n}. This way we have Q \subseteq \Re. We have thus effectively extended to \Re.

More later,

Nalin Pithwa