Question

Test the series for convergence or divergence

Answer 1

\[\sum_{n=1}^{\infty}\frac{ 1 }{ 2^{n}-1 }\]

Answer 2

converges like there is not tomorrow

Answer 3

*no tomorrow
compare it to the geometric series
\[\sum\frac{1}{2^n}=1\]

Answer 4

i need to test it by the limit comparison test

Answer 5

yes
because \(2^n>2^n-1\) so
\[\frac{1}{2^n}<\frac{1}{2^n-1}\] so you do need the limit comparison test

Answer 6

but since
\[\lim_{n\to \infty}\frac{2^n}{2^n-1}=1\] you win right away