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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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