Let

Express as an infinite series, find the sum, and express as a quotient of two integers.

We have

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#
Stumbling Robot

A Fraction of a Dot
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Express the decimal *x = 0.51515151…* as a quotient of two integers

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### Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):

Let

Express as an infinite series, find the sum, and express as a quotient of two integers.

We have

The definition provided by Apostol means that the sum is not 51/100+51/100^2+… but 0+5/10+1/10^2 +5/10^3 +… which I solved by taking the partial sum from k=0 to 2n of ak where a0=0 and an=(3+2(-1)^n+1)/10^n for n greater than 0. The next few exercises are trickier and require defining the terms using mod(p) notation and taking the partial sum up to np.