Consider the sequence \(a_n\) whose \(n\)th term is given by the reciprocal of the first digit of \(n!\). For example, \(a_1=1\) since the first digit of \(1!=1\) is \(1\); \(a_6=\dfrac{1}{7}\) since \(6!=720\), and so on.
Take another sequence, \(b_n\), a subsequence of \(a_n\) that doesn't contain \(1\). (This measure is taken just in case there are infinitely many terms of \(a_n\) that are \(1\).)
Does \(\displaystyle\sum_{n=1}^\infty b_n\) converge?

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darn it always get confused about 1st and last
we are definitely not looking at the units digit

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