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In [[mathematics]], a '''rational zeta series''' is the representation of an arbitrary [[real number]] in terms of a series consisting of [[rational number]]s and the [[Riemann zeta function]] or the [[Hurwitz zeta function]]. Specifically, given a real number ''x'', the rational zeta series for ''x'' is given by
And keep away from namkeens, junk and oily food from outside which are the sources of unhealthy fats. The Paleo Diet is one of the most popular diets in today's generation. Watch out for them, especially when you're purchasing nuts, and other items that have been packaged. Delicious Paleo Brunch, Lunch and Dinner Salad Recipe. Many vitamins and medications use gluten as a binding agent. <br><br>So specifically exactly what really does one particular distinct eat and also by no implies eat right after getting Paleo Recipe Book. You don't have to buy special mail-order food that costs a fortune. Ready-made paleo foods items are starting to find their way into conventional grocery stores. In addition, food is something that we can't turn down yet healthy foods is another matter to discuss. A diet that is rich with calcium may help to prevent tooth decay, but when a diet is low in calcium, as a majority of American diets are, the body takes that mineral from teeth and bones. <br><br>James, who famously returned to the Cleveland Cavaliers, is following in the fork-steps of his buddy, Miami Heat point guard Dwyane Wade, who has been adhering to the Paleo plan on the advice of fellow Heat player Ray Allen, who has been eating similarly since 2013. "That, and realizing you're gonna lose everything you have. Those that follow this program can expect a lot of red meat as well as fatty food. - Saut chicken cutlets on both sides until thoroughly cooked. Paleo recipes and the lifestyle that is associated with them have been steadily growing in popularity. <br><br>DASH stands for Dietary Approaches to Stop Hypertension, but it can help with much more than that. Remove the washcloth and use it to gently wipe off the oil. a Mediterranean style of eating combined with physical activity is the optimal lifestyle plan for preventing a second heart attack. This is the workhorse phase of the diet, when you flip the pure protein days of Attack with days when you bring a delightful array of limitless salads and vegetables to your wide choice of meat, fish and no-fat milk products. If you are really committed to the Paleo Diet plan, then gone are the times of buying for "staples" like bread and milk. <br><br>These tips will help you initially navigate your new eating habits. This diet, also known as the caveman diet, stone age diet, or hunter-gatherer diet, is based on  the habitually food consumption patterns of humans during the Paleolithic era. It is infinitely customizable to meet YOUR specific needs and goals. Protein satisfies your appetite more than any other macronutrient, it increases metabolism, and it will contribute to weight loss. Whereas before the author strictly advocated following the original Paleo guidelines, this new version allows the addition of dairy products especially for athletes who want to build muscle and gain weight. <br><br>That’s more than are affected by Multiple Sclerosis, Crohn’s Disease, Cystic Fibrosis, and Parkinson’s combined. Here are some items that you will surely love about this. And that philosophy definitely applies to the Paleo low carb diet, based on the theory that we can thrive by mirroring the diets of our hunter-gatherer ancestors. A paleo breakfast is protein and fat abundant, carbohydrate poor. Some of the most devoted Paleolithic adherents also adopt a workout regimen that is thought to be based more closely on the workouts of our ancestors.<br><br>If you liked this article so you would like to get more info relating to [http://gritsandgroceries.info/sitemap/ paleo brownies recipe] i implore you to visit our web site.
 
:<math>x=\sum_{n=2}^\infty q_n \zeta (n,m)</math>
 
where ''q''<sub>''n''</sub> is a rational number, the value ''m'' is held fixed, and &zeta;(''s'',&nbsp;''m'') is the Hurwitz zeta function. It is not hard to show that any real number ''x'' can be expanded in this way.
 
==Elementary series==
For integer ''m>1'', one has
 
:<math>x=\sum_{n=2}^\infty q_n \left[\zeta(n)- \sum_{k=1}^{m-1} k^{-n}\right] </math>
 
For ''m=2'', a number of interesting numbers have a simple expression as rational zeta series:
 
:<math>1=\sum_{n=2}^\infty \left[\zeta(n)-1\right]</math>
 
and
:<math>1-\gamma=\sum_{n=2}^\infty \frac{1}{n}\left[\zeta(n)-1\right]</math>
 
where &gamma; is the [[Euler–Mascheroni constant]].  The series
:<math>\log 2 =\sum_{n=1}^\infty \frac{1}{n}\left[\zeta(2n)-1\right]</math>
 
follows by summing the [[Gauss–Kuzmin distribution]]. There are also series for &pi;:
 
:<math>\log \pi =\sum_{n=2}^\infty \frac{2(3/2)^n-3}{n}\left[\zeta(n)-1\right]</math>
 
and
 
:<math>\frac{13}{30} - \frac{\pi}{8} =\sum_{n=1}^\infty \frac{1}{4^{2n}}\left[\zeta(2n)-1\right]</math>
 
being notable because of its fast convergence. This last series follows from the general identity
 
:<math>\sum_{n=1}^\infty (-1)^{n} t^{2n} \left[\zeta(2n)-1\right] =
\frac{t^2}{1+t^2} + \frac{1-\pi t}{2} - \frac {\pi t}{e^{2\pi t} -1} </math>
 
which in turn follows from the [[generating function]] for the [[Bernoulli numbers]]
 
:<math>\frac{x}{e^x-1} = \sum_{n=0}^\infty B_n \frac{t^n}{n!}</math>
 
Adamchik and Srivastava give a similar series
 
:<math>\sum_{n=1}^\infty \frac{t^{2n}}{n} \zeta(2n) =
\log \left(\frac{\pi t} {\sin (\pi t)}\right)</math>
 
==Polygamma-related series==
A number of additional relationships can be derived from the [[Taylor series]] for the [[polygamma function]] at ''z''&nbsp;=&nbsp;1, which is  
:<math>\psi^{(m)}(z+1)= \sum_{k=0}^\infty
(-1)^{m+k+1} (m+k)!\; \zeta (m+k+1)\; \frac {z^k}{k!}</math>.
The above converges for |''z''|&nbsp;&lt;&nbsp;1. A special case is
 
:<math>\sum_{n=2}^\infty t^n \left[\zeta(n)-1\right] =
-t\left[\gamma +\psi(1-t) -\frac{t}{1-t}\right]
</math>
 
which holds for |''t''|&nbsp;&lt;&nbsp;2. Here, &psi; is the [[digamma function]] and &psi;<sup>(''m'')</sup> is the polygamma function. Many series involving the [[binomial coefficient]] may be derived:
 
:<math>\sum_{k=0}^\infty {k+\nu+1 \choose k} \left[\zeta(k+\nu+2)-1\right]
= \zeta(\nu+2)</math>
 
where &nu; is a complex number. The above follows from the series expansion for the Hurwitz zeta
 
:<math>\zeta(s,x+y) =
\sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x)</math>
taken at ''y''&nbsp;=&nbsp;&minus;1. Similar series may be obtained by simple algebra:
 
:<math>\sum_{k=0}^\infty {k+\nu+1 \choose k+1} \left[\zeta(k+\nu+2)-1\right]
= 1</math>
 
and
 
:<math>\sum_{k=0}^\infty (-1)^k {k+\nu+1 \choose k+1} \left[\zeta(k+\nu+2)-1\right]
= 2^{-(\nu+1)} </math>
 
and
 
:<math>\sum_{k=0}^\infty (-1)^k {k+\nu+1 \choose k+2} \left[\zeta(k+\nu+2)-1\right]
= \nu \left[\zeta(\nu+1)-1\right] -  2^{-\nu}</math>
and
 
:<math>\sum_{k=0}^\infty (-1)^k {k+\nu+1 \choose k} \left[\zeta(k+\nu+2)-1\right]
= \zeta(\nu+2)-1 -  2^{-(\nu+2)}</math>
 
For integer ''n''&nbsp;&ge;&nbsp;0, the series
:<math>S_n = \sum_{k=0}^\infty {k+n \choose k} \left[\zeta(k+n+2)-1\right]</math>
 
can be written as the finite sum
 
:<math>S_n=(-1)^n\left[1+\sum_{k=1}^n \zeta(k+1) \right] </math>
 
The above follows from the simple [[recursion relation]] ''S''<sub>''n''</sub>&nbsp;+&nbsp;''S''<sub>''n''&nbsp;+&nbsp;1</sub>&nbsp;=&nbsp;&zeta;(''n''&nbsp;+&nbsp;2).  Next, the series
 
:<math>T_n = \sum_{k=0}^\infty {k+n-1 \choose k} \left[\zeta(k+n+2)-1\right]</math>
 
may be written as
 
:<math>T_n=(-1)^{n+1}\left[n+1-\zeta(2)+\sum_{k=1}^{n-1} (-1)^k (n-k) \zeta(k+1) \right] </math>
 
for integer ''n''&nbsp;&ge;&nbsp;1. The above follows from the identity ''T''<sub>''n''</sub>&nbsp;+&nbsp;''T''<sub>''n''&nbsp;+&nbsp;1</sub>&nbsp;=&nbsp;''S''<sub>''n''</sub>.  This process may be applied recursively to obtain finite series for general expressions of the form
 
:<math>\sum_{k=0}^\infty {k+n-m \choose k} \left[\zeta(k+n+2)-1\right]</math>
 
for positive integers ''m''.
 
==Half-integer power series==
Similar series may be obtained by exploring the [[Hurwitz zeta function]] at half-integer values. Thus, for example, one has
 
:<math>\sum_{k=0}^\infty \frac {\zeta(k+n+2)-1}{2^k}
{{n+k+1} \choose {n+1}}=\left(2^{n+2}-1\right)\zeta(n+2)-1</math>
 
==Expressions in the form of p-series==
Adamchik and Srivastava give
:<math>\sum_{n=2}^\infty n^m \left[\zeta(n)-1\right] =
1\, +
\sum_{k=1}^m k!\; S(m+1,k+1) \zeta(k+1)</math>
 
and
 
:<math>\sum_{n=2}^\infty (-1)^n n^m \left[\zeta(n)-1\right] =
-1\, +\, \frac {1-2^{m+1}}{m+1} B_{m+1}
\,- \sum_{k=1}^m (-1)^k k!\; S(m+1,k+1) \zeta(k+1)</math>
 
where <math>B_k</math> are the [[Bernoulli number]]s and <math>S(m,k)</math> are the [[Stirling numbers of the second kind]].
 
==Other series==
Other constants that have notable rational zeta series are:
* [[Khinchin's constant]]
* [[Apéry's constant]]
 
==References==
* {{cite journal|author=Jonathan M. Borwein, David M. Bradley, Richard E. Crandall
|title=Computational Strategies for the Riemann Zeta Function
|journal=J. Comp. App. Math.
|year=2000
|volume=121
|pages=p.11
|url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf}}
* {{cite journal|author=Victor S. Adamchik and H. M. Srivastava
|title=Some series of the zeta and related functions
|journal=Analysis
|year=1998
|volume=18
|pages=pp. 131–144
|url=http://www-2.cs.cmu.edu/~adamchik/articles/sums/zeta.pdf}}
 
[[Category:Zeta and L-functions]]
[[Category:Real numbers]]

Latest revision as of 09:06, 22 May 2014

And keep away from namkeens, junk and oily food from outside which are the sources of unhealthy fats. The Paleo Diet is one of the most popular diets in today's generation. Watch out for them, especially when you're purchasing nuts, and other items that have been packaged. Delicious Paleo Brunch, Lunch and Dinner Salad Recipe. Many vitamins and medications use gluten as a binding agent.

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