[See Steps] An integer is said to be 8 -free if it does not contain the digit 8 in its decimal expansion. So 17 is 8 -free, but 1081 is not. Denote the set
Question: An integer is said to be 8 -free if it does not contain the digit 8 in its decimal expansion. So 17 is 8 -free, but 1081 is not. Denote the set of 8 -free integers by \(\mathcal{S}\). Prove that the series
\[\sum_{n \in \mathcal{S}} \frac{1}{n}=1+\frac{1}{2}+\cdots+\frac{1}{7}+\frac{1}{9}+\cdots+\frac{1}{17}+\frac{1}{19}+\cdots\]is convergent. (Hint: Work out how many reciprocals of one digit 8 -free numbers there are in the sum, how many reciprocals of 2 digit 8 -free numbers, 3 digit 8 -free etc, then bound each group of k-digit terms.)
Deliverable: Word Document 
![[Solution Library] (a) Find the Taylor series](/images/solutions/MC-solution-library-38689.jpg "[Solution Library] (a) Find the Taylor series expansion for y= sin")
[Solution Library] (a) Find the Taylor series expansion for y= sin
Solution: Find the Fourier cosine series for f(t)=t^2, 0 ∑_n=1^∞
Solution: Solve the Laplace equation with the following boundary
Solution: Solve the equation u_t=u_x x+c u_x, for 0 ≤q x ≤q
![[See Solution] Show that K(x, t)=(1)/(√4 π](/images/solutions/MC-solution-library-38693.jpg "[See Solution] Show that K(x, t)=(1)/(√4 π t) e^-x^2 /")
[See Solution] Show that K(x, t)=(1)/(√4 π t) e^-x^2 /
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