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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 Sums

         

What is the value of \[\sum_{n=0}^{\infty} \left(2 ^{n+1}-1\right) \left(\frac{1}{9}\right)^{n}?\]

Determine the sum \[\sum_{n=1}^{\infty} 63 ^{n-1} \left(\frac{1}{8}\right)^{2n}.\]

What is the value of \[\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n \left(\frac{5}{4}\right)^{2n}?\]

If \(m\) is a positive integer such that \[\sum_{n=1}^{\infty} \log \left(1-\frac{1}{(n+5)^2}\right) = \log \frac{m}{m+1},\] what is the value of \(m?\)

What is the value of \[\sum_{n=0}^{\infty} \left(3 ^{n+1}-1\right) \left(\frac{1}{8}\right)^{n}?\]

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