Express as an infinite series, find the sum, and express as a quotient of two integers.
We have
Related
One comment
Mohammad Azad says:
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.
Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): Cancel reply
![\[ x = 0.51515151\ldots. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-344b9143461dea8e1275b6daf4e93fea_l3.png "Rendered by QuickLaTeX.com")
as an infinite series, find the sum, and express ![\begin{align*} x = 0.515151\ldots && \implies && x &= \sum_{k=1}^{\infty} \frac{51}{100^k} \\[9pt] &&&&&= 51 \sum_{k=0}^{\infty} \left( \frac{1}{100} \right)^k - 51 \\[9pt] &&&&&= 51 \left( \frac{1}{1-\frac{1}{100}} \right) - 51 \\[9pt] &&&&&= \frac{51}{99}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-14900450d0aa53c72afeaaa6f29c2b47_l3.png "Rendered by QuickLaTeX.com")
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.