What is some infinite series expansion for 3/7? and so on for these fraction with digit in base 10? I can't think of some useful thing at all. Please generalize some useful series expression for all those kinds of fraction.
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One might also try the mundane $$\frac{3}{7} = \frac{1}{{2 + \frac{1}{3}}} = \frac{1}{2}\frac{1}{{1 + \frac{1}{6}}} = \frac{1}{2}\left( {1 - \frac{1}{6} + \frac{1}{{{6^2}}} - \frac{1}{{{6^3}}} + - \cdots } \right)$$ |
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Consider any convergent series. Lets say that $$a_1 + a_2 + a_3 + \cdots = a$$ where $a_n, a \in \mathbb{R}$ and $a \neq 0$ (Thanks, @anon). Define $b_n = \dfrac37\dfrac{a_n}{a}$, then the series $b_1 + b_2 + b_3 + \cdots$ converges to $\dfrac37$. |
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Find the smallest $n$ such that $7|(10^n-1)$. In particular say $10^n-1=7m$. Then write $$\begin{array}{c l} \frac{3}{7} & =\frac{3m}{7m} \\[2pt] & =\frac{3m}{10^n-1} \\[2pt] & = 3m\frac{1}{10^n}\frac{1}{1-1/10^n} \\[2pt] & =\frac{3m}{10^n}+\frac{3m}{10^{2n}}+\frac{3m}{10^{3m}}+\cdots. \end{array}$$ Since $3m<7m+1=10^n$, this gives an easy way to write out the repeating decimal expansion of the fraction $3/7$. In particular, $10^6-1=7\cdot142857$, and $3\cdot142857=428571$ so $3/7=0.\overline{428571}$. This can be employed more generally. Without much loss of generality assume $0<a<b$ with $b$ divisible by neither $2$ nor $5$ and find an $n$ such that $b|(10^n-1)$, in particular $10^n-1=bm$, so that $$\frac{a}{b}=\frac{am}{10^n}+\frac{am}{10^{2n}}+\frac{am}{10^{3m}}+\cdots.$$ Note again $am<bm+1=10^n$ so this provides a clean decimal expansion. We ignore integers, and for rationals more generally we can decompose them into a sum of the nearest integer plus the rational's fractional part, and apply this method to the fractional part, then recombine. For rationals with $2,5$'s in the denominator's prime factorization, pull them out and multiply by them afterwards: their reciprocals are simply $0.5$ and $0.2$ after all. Finally, for base $r$ (assuming $r$ is a positive natural number anyway), find $n$ such that $b|(r^n-1)$ and adapt this method. (Remember to pull out $\gcd(b,r)$ from the denominator.) Note that $n$ is guaranteed to exist because $r$ has an order modulo $b'$ ($b'$ coprime) by elementary number theory. Also, if $d|r$, then $\frac{1}{d}=\frac{(r/d)}{r}$ is an easy way to compute the base-$r$ expansion of $1/d$. This helps with the "pulled-out" factors (i.e. with $d=\gcd(b,r)$). |
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Using the formula for the sum of a geometric series, we get $$ \sum_{k=1}^\infty\frac{3}{8^k}=\frac38\sum_{k=0}^\infty\frac{1}{8^k}=\frac38\frac{1}{1-\frac18}=\frac38\frac87=\frac37 $$ Therefore, $$ \frac37=\sum_{k=1}^\infty\frac{3}{8^k} $$ |
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Infinite series expansion for a number? There are many series whose is $\,3/7\,$, some of them rather dandy: $$\frac{1}{2}\sum^\infty_{n=0}\frac{1}{(-6)^n}$$$$\frac{18}{7\pi^2}\sum^\infty_{n=1}\frac{1}{n^2}$$ and a long etc. |
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How about a few telescoping series? $$\begin{align*} \frac37&=\sum_{k=1}^\infty\frac{40}{(7k+21)(3k+15)}\\ &=\sum_{k=1}^\infty\frac{100}{(17k+34)(3k+21)}\\ &=\sum_{k=1}^\infty\frac{36}{(k+5)(k+6)(k+7)}\\ &=\sum_{k=1}^\infty\frac{48}{(k+6)(k+7)(k+8)} \end{align*}$$ |
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