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mesa
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The title pretty much says it all, does anyone know infinite series summations that are equal to 1/5 or 1/7?
AlephZero said:You can easily invent a geometric series that sums to any value you want.
Or you could do something like the series expansion of ##\frac 1 5 \sin \pi##.
That won't get him 1/5, though. :tongue:AlephZero said:Or you could do something like the series expansion of ##\frac 1 5 \sin \pi##.
Indeed.jgens said:How about the trivial example where the first term in the series is 1/5 and all the remaining terms are zero? No need to get all fancy here guys!
Office_Shredder said:Any function whose value you can calculate and whose Taylor series can be expanded will give you a series. What is the purpose of this question?
Mandelbroth said:That won't get him 1/5, though. :tongue:
I'm sure that was a typo. mesa, he probably meant ##\frac{\sin\frac{\pi}{2}}{5}##.
jgens said:How about the trivial example where the first term in the series is 1/5 and all the remaining terms are zero? No need to get all fancy here guys!
Mandelbroth said:Indeed.
mesa, you should look back at your other thread on infinite series. Many of the same series (plural) apply here.
Borek said:Get any sum that yields any finite number, multiply it by a "normalizing factor".
[tex]e=\sum_0^\infty\frac 1 {n!}[/tex]
so
[tex]\frac 1 5 = \sum_0^\infty\frac 1 {5en!}[/tex]
Seydlitz said:One mentor gave me this link a couple of months ago, it's basically a comprehensive table of integrals and sum of series. You can find pretty interesting series there.
http://atsol.fis.ucv.cl/dariop/sites...ij_Engl._2.pdf
Mandelbroth said:I'm sure that was a typo.
Nah. We all make mistakes or have lapses in genius from time to time. It's all good.AlephZero said:More of a brain-fart than a typo. I divided ##2\pi## by ##2## and got ##\pi##. Full marks for arithmetic, but zero marks for doing the wrong sum.
What do you mean "more general"?mesa said:Yes they do, I was curious if we had more 'general' solutions for infinite series that would give whole integer fractions.
Mandelbroth said:What do you mean "more general"?
Robert1986 said:If you want a series that converges to c, take a series that converges to b, and add this term: (c-b) to the front.
mesa said:That certainly would work much like Boreks suggestion of using a 'normalizing factor'.
Do you know of any series that will equal all whole integer fractions by simply changing the lower limit of summation?
Mandelbroth said:Consider the sequence $$a_i=\{1,-1,\frac{1}{2},\frac{1}{2},-1,\frac{1}{3},\frac{1}{3},\frac{1}{3},-1,...\}.$$
##\displaystyle \sum_{n_0\leq i\leq n}a_i## should cover everything you want. :tongue:
An infinite series summation is a mathematical expression that represents the sum of an infinite number of terms. It is written in the form of ∑n=1∞ an, where an is the nth term of the series.
Yes, an infinite series summation can be equal to 1/5 or 1/7. It depends on the specific series and the values of its terms.
An infinite series summation being equal to 1/5 or 1/7 can have different mathematical implications depending on the context. It could represent a specific value or convergence of a series, which can be used in various mathematical calculations and formulas.
There is no one specific method for finding an infinite series summation that is equal to 1/5 or 1/7. It requires knowledge of mathematical concepts and techniques such as convergence tests, geometric series, and telescoping series.
One example of an infinite series summation that is equal to 1/5 is the series ∑n=1∞ 1/(5n). This series is a geometric series with a common ratio of 1/5, and it converges to 1/4. An example of an infinite series summation that is equal to 1/7 is the series ∑n=1∞ 1/(7n). This series is also a geometric series with a common ratio of 1/7, and it converges to 1/6.