Limits of Sequences and Series

Infinite Sums


What is the value of n=0(2n+11)(19)n?\sum_{n=0}^{\infty} \left(2 ^{n+1}-1\right) \left(\frac{1}{9}\right)^{n}?

Determine the sum n=163n1(18)2n.\sum_{n=1}^{\infty} 63 ^{n-1} \left(\frac{1}{8}\right)^{2n}.

What is the value of n=1(12)n(54)2n?\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n \left(\frac{5}{4}\right)^{2n}?

If mm is a positive integer such that n=1log(11(n+5)2)=logmm+1,\sum_{n=1}^{\infty} \log \left(1-\frac{1}{(n+5)^2}\right) = \log \frac{m}{m+1}, what is the value of m?m?

What is the value of n=0(3n+11)(18)n?\sum_{n=0}^{\infty} \left(3 ^{n+1}-1\right) \left(\frac{1}{8}\right)^{n}?


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