The Pension Fund Puzzle

The Story: Radhika is a financial analyst at a small pension fund office in Pune. Every quarter, she calculates how much money the fund will have after a fixed period of time, given a starting amount and a fixed annual interest rate. One afternoon, the fund's director walks in with an unusual question. 'I have a thought experiment,' he says. 'Suppose we invest one rupee at an annual interest rate of 100%, compounded continuously — not once a year, but infinitely many times per year. What does the investment grow to after exactly one year?' Radhika smiles. 'That is a famous limit,' she says. 'It is connected to the most important constant in all of mathematics.' The director raises an eyebrow. 'And what does it equal, exactly?' Radhika writes a number on her notepad and slides it across the desk. The Challenge: A sum of money is invested at 100% annual interest rate, compounded n times per year. Write the formula for the value after 1 year. Find the limiting value as n approaches infinity. What famous constant does this produce, and what is its approximate numerical value?