Calculus
# Sequences and Series

How many squares are present in figure number 20?

When I was a teenager, a friend and I stole soda from our parents. I took the soda bottle, drank half of what's there, and then passed it to my friend. My friend drank half of what was left, and passed it back to me. I again drank half of what was left, and passed it back to him.

We kept up with this until the entire soda bottle was empty.

Approximately what percentage of the soda did I drink?

\[ \frac{100001+100003+100005+\cdots+199999}{1+3+5+7+\cdots+99999} = \ ? \]

\[\left(\dfrac1{2^2}+\dfrac1{3^2}+\dfrac1{4^2}+\ldots\right) + \\ \hspace{1cm}\left(\dfrac1{2^3}+\dfrac1{3^3}+\dfrac1{4^3} + \ldots\right) + \\ \hspace{2.5cm}\left(\dfrac1{2^4}+\dfrac1{3^4}+\dfrac1{4^4}+\ldots\right)+\cdots\]

What is the value of the series above?

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