Stock Converges to Zero
The price of a stock starts at $100, then drops 1%, then grows 1%, then drops 1% again and so on. What does the price converge to?
Answer
Every “down 1% then up 1%” cycle multiplies the price by
\[
\frac{99}{100}\times\frac{101}{100}
\;=\;\frac{9999}{10000}
\;<\;1.
\]
Hence after \(k\) cycles the price is
\[
100\Bigl(\tfrac{9999}{10000}\Bigr)^k,
\]
and as \(k\to\infty\) this tends to \(0\).