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## Limits of Sequences and Series

Infinitely many mathematicians walk into a bar. The first says "I'll have a beer". The next ones say "I'll have half of the previous guy". The bartender pours out 2 beers and says "Know your limits".

# Infinite Series Warmup

$\sum_{n=1}^{\infty}3^n$

Does this geometric series converge to a finite number?

$\sum_{n=1}^{\infty}\left( \frac{1}{3} \right) ^n$

Does this geometric series converge to a finite number?

To what value does $\sum_{n=1}^{\infty} 5 \left( \frac{1}{5} \right) ^{n-1}$

converge?

Consider the series:

$\text{A. }\sum_{n=1}^{\infty} \frac{1}{n^2}$

$\text{B. }\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$

$\sum_{n=1}^{\infty} (-1)^n$

Does this series converge?

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