Calculus
# Sequences and Series

True or False?

For every quadratic function \(f(x) = ax^2 + bx + c,\) there is a real number \(x\) such that \[f(x)=f'(x).\]

What is the sum of the areas of the removed squares if this process continues infinitely?

Evaluate \[\int_{0}^{3} \lfloor x \rfloor dx.\]

**Notation.**\(\lfloor x \rfloor\) represents the *floor,* or integer portion of \(x.\) For example, \(\lfloor 5.81 \rfloor = 5.\)

For what positive integer \(n\) is \[\int_0^n \lfloor x \rfloor dx = \lfloor \int_0^n x dx \rfloor ?\]

**Notation.**\(\lfloor x \rfloor\) represents the *floor,* or integer portion of \(x.\) For example, \(\lfloor 5.81 \rfloor = 5.\)

True or False?

For every \(a > 2,\)

\[\sum_{n=0}^{\infty}\frac{1}{a^n} < \sum_{n=0}^{\infty}\frac{1}{n!}.\]

Hint. \[\sum_{n=0}^{\infty}\frac{1}{n!} = e.\]

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