Let a_n be the n_th positive integer whose decimal representation does not contain the digit 9. Does the series ∑(n = 1, ∞) 1/a_n converge or diverge?
Thank you.
Thank you.
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The sum converges.
The number c(n) of integers with n-digit decimal expansions avoiding the digit 9 is
c(n) = 8·9ⁿ⁻¹
because the first digit is in {1,2,3,⋯,8} and the last n-1 digits are all in {0,1,2,⋯,8}.
Now
∑1/a_i
= ∑_[n≥1] ∑_[j such that the expansion of j avoids 9 and has n digits] 1/j
≤ ∑_[n≥1] ∑_[j such that the expansion of j avoids 9 and has n digits] 1/10ⁿ⁻¹
= ∑_[n≥1] 8·9ⁿ⁻¹·1/10ⁿ⁻¹
= 8·∑_[n≥1] 9ⁿ⁻¹/10ⁿ⁻¹
= 8·∑_[n≥0] 9ⁿ/10ⁿ
= 8·1/(1 - 9/10) = 80.
(The inequality follows from 10ⁿ⁻¹ ≤ j, for any integer j with an n-digit decimal expansion).
The number c(n) of integers with n-digit decimal expansions avoiding the digit 9 is
c(n) = 8·9ⁿ⁻¹
because the first digit is in {1,2,3,⋯,8} and the last n-1 digits are all in {0,1,2,⋯,8}.
Now
∑1/a_i
= ∑_[n≥1] ∑_[j such that the expansion of j avoids 9 and has n digits] 1/j
≤ ∑_[n≥1] ∑_[j such that the expansion of j avoids 9 and has n digits] 1/10ⁿ⁻¹
= ∑_[n≥1] 8·9ⁿ⁻¹·1/10ⁿ⁻¹
= 8·∑_[n≥1] 9ⁿ⁻¹/10ⁿ⁻¹
= 8·∑_[n≥0] 9ⁿ/10ⁿ
= 8·1/(1 - 9/10) = 80.
(The inequality follows from 10ⁿ⁻¹ ≤ j, for any integer j with an n-digit decimal expansion).
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Diverge