Let a1; a2; a3;::: be those positive integers that do not co
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Let a1; a2; a3;::: be those positive integers that do not contain the digit 0 in their decimal representations. Thus, a1
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Let a1; a2; a3;::: be those positive integers that do not contain the digit 0 in their decimal representations. Thus, a1 D 1, a2 D 2, :::; a9 D 9, a10 D 11, :::; a18 D 19, a19 D 21, :::; a90 D 99, a91 D 111, etc. Show that the series X1 nD1 1 an converges and that the sum is less than 90. (Hint: How many of these integers have m digits? Each term 1=an, where an has m digits, is less than 10?mC1.)
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