Calculus

# 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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