Inelegant Explorations of Some Personal Problems

A Simple Method to Compute the Sum of Powers of Natural Numbers

One fine summer morning not so long ago, I woke up with a start to the stark realization that I had no idea how to derive the formula for the sum of powers of natural numbers. I am talking about a closed-form expression for the series (for non-negative integer $p$) $\sum_{i=1}^n {i^p} = 1 + {2^p} + {3^p} + \dots + {n^p}$

Estimating Expectations and Variances using Python

Last week, for pedagogical reasons, I had to solve the following problem: In a bag there are $4$ blue and $5$ red balls. We randomly take $3$ balls from the bag without replacement. Let the random variable $X$ denote the number of blue balls we get. Find out the expectation and variance of $X$.

Harmonic Integers

Years back, I encountered the following problem in a book on Number Theory: Prove that the sum $H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ is never an integer after $n > 1$. In other words, in the sequence $1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \dots$, there is only one integer, and that is the first number $1$.

A Triangle from a Stick

A famous problem in probability is this: Suppose you randomly1 break a stick in two places. You will get three smaller pieces of stick. What is the probability that you can form a triangle using these pieces? One interesting thing about this problem is that depending on how you visualize breaking the stick, you get different probabilities.