Discussion:
infinite sum
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d***@gmail.com
2017-06-16 02:07:56 UTC
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what do infinite sum x^k/(3k+2) as k ->infinity equal to?

reference math handbook
www.mathHandbook.com
William Elliot
2017-06-16 02:37:52 UTC
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Post by d***@gmail.com
what do infinite sum x^k/(3k+2) as k ->infinity equal to?
Huh? Are you asking if sum(k=1,oo) x^k/(3k + 2) converges?
d***@gmail.com
2017-06-16 05:05:49 UTC
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Post by William Elliot
Post by d***@gmail.com
what do infinite sum x^k/(3k+2) as k ->infinity equal to?
Huh? Are you asking if sum(k=1,oo) x^k/(3k + 2) converges?
yes. what is euqal to =?

reference math handbook
www.mathHandbook.com
m***@wp.pl
2017-06-16 08:58:42 UTC
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Post by d***@gmail.com
Post by William Elliot
Post by d***@gmail.com
what do infinite sum x^k/(3k+2) as k ->infinity equal to?
Huh? Are you asking if sum(k=1,oo) x^k/(3k + 2) converges?
yes. what is euqal to =?
It's aleph0. Sorry to say, but if You have to ask
such things you should avoid mathematical problems.
m***@wp.pl
2017-06-16 09:03:27 UTC
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Post by m***@wp.pl
Post by d***@gmail.com
Post by William Elliot
Post by d***@gmail.com
what do infinite sum x^k/(3k+2) as k ->infinity equal to?
Huh? Are you asking if sum(k=1,oo) x^k/(3k + 2) converges?
yes. what is euqal to =?
It's aleph0. Sorry to say, but if You have to ask
such things you should avoid mathematical problems.
A nice compromitation for me. Sorry for that.
David Petry
2017-06-16 08:22:22 UTC
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Post by d***@gmail.com
what do infinite sum x^k/(3k+2) as k ->infinity equal to?
reference math handbook
www.mathHandbook.com
WolframAlpha gives the answer in terms of hypergeometric functions. Doing it by hand gives an answer in terms of logarithms.

Does your "mathHandbook" answer the question?
d***@gmail.com
2017-06-17 00:15:52 UTC
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Post by David Petry
Post by d***@gmail.com
what do infinite sum x^k/(3k+2) as k ->infinity equal to?
reference math handbook
www.mathHandbook.com
WolframAlpha gives the answer in terms of hypergeometric functions.
it is too complicated. there is a simple result but wolfram cannot do it.
Post by David Petry
Doing it by hand gives an answer in terms of logarithms.
what is your result?
Post by David Petry
Does your "mathHandbook" answer the question?
input your formula
x^k/(3k+2)
into website www.mathHandbook.com, click the sum button for answer.
http://www.mathhandbook.com/input/?guess=infsum%28x%5Ek%2F%283k%2B2%29%29&inp=x%5Ek%2F%283k%2B2%29


reference math handbook
www.mathHandbook.com
Roland Franzius
2017-06-17 05:59:21 UTC
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Post by d***@gmail.com
Post by David Petry
Post by d***@gmail.com
what do infinite sum x^k/(3k+2) as k ->infinity equal to?
reference math handbook
www.mathHandbook.com
WolframAlpha gives the answer in terms of hypergeometric functions.
it is too complicated. there is a simple result but wolfram cannot do it.
Post by David Petry
Doing it by hand gives an answer in terms of logarithms.
what is your result?
Post by David Petry
Does your "mathHandbook" answer the question?
input your formula
x^k/(3k+2)
into website www.mathHandbook.com, click the sum button for answer.
http://www.mathhandbook.com/input/?guess=infsum%28x%5Ek%2F%283k%2B2%29%29&inp=x%5Ek%2F%283k%2B2%29
This gives the most superflous answer a computer can give in such a
case. Any Taylor series Phi(x,...) = sum_k a_k x^k with a simple
multiply fractional linear regression formula of the coefficients a_k->
a_(k+1) is called a Gauss hypergeometric function as a generalization of
the geometric series (1-x)^-1 = sum_k x^k.

So, what is gained to display a clear and simple infinite sum as a
member of the family of hypergeometric functions with a special
selection of its many parameters.

(answer: a one-liner in Mathematica eg to retrieve the coeffients in the
recursion formula for the coefients. Nobody with the least mathematical
skill uses a computer algebra system for such a nonsense)

This is the "Schwachsinn"
(http://dict.leo.org/englisch-deutsch/schwachsinn) of the form we mostly
find in help files from other idiots in the computer and social sciences
branches, that explicate unknown terms in contexts of more
even-more-unknown terms.
--
Roland Franzius
Markus Klyver
2017-06-16 09:01:12 UTC
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It will only converge if abs(x) =< 1.
Jim Burns
2017-06-16 12:39:56 UTC
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Post by d***@gmail.com
what do infinite sum x^k/(3k+2) as k ->infinity equal to?
I'll call that sum S, assuming k >= 0.

Let x = y^3.
The sum S
1/2 + x/5 + x^2/8 + ...
equals
(1/y^2)( y^2/2 + y^5/5 + y^8/8 + ... )

Note that
y/1 + y^2/2 + y^3/3 + y^4/4 + y^5/5 + ...
equals
ln(1/(1 - y))

for |y| =< 1 & ~(y = 1)

Define a clock sequence c(k) such that
c(1), c(2), c(3), c(4), c(5), c(6), ...
equals
0, 1, 0, 0, 1, 0, ...

The sum
c(1)*y/1 + c(2)*y^2/2 + c(3)*y^3/3 + ...
equals
y^2/2 + y^5/5 + y^8/8 + ...

Let e be a solution of u^3 = 1 other than u = 1.
Thus
(e^3 - 1)(e - 1) = e^2 + e + 1 = 0
There are two possible solution. Let
e = - 1/2 + sqrt(3)/2*i
Define
c(k-1) = ( e^2k + e^k + 1 )/3
Then
c(1), c(2), c(3), c(4), c(5), c(6), ...
equals
0, 1, 0, 0, 1, 0, ...

If the following converges absolutely, we can
re-arrange the terms of
c(1)*y/1 + c(2)*y^2/2 + c(3)*y^3/3 + ...
as the sum of three infinite sums
(e^2/3)( (e^2*y)/1 + (e^2*y)^2/2 + ... )
+
(e/3)( (e*y)/1 + (e*y)^2/2 + (e*y)^3/3 + ... )
+
(1/3)( y/1 + y^2/2 + y^3/3 + y^4/4 + ... )

which equals
(e^2/3)*ln(1/(1 - e^2*y))
+ (e/3)*ln(1/(1 - e*y))
+ (1/3)*ln(1/(1 - y))

for |y| < 1 and e = - 1/2 + sqrt(3)/2*i

I will write cbrt(x) = x^(1/3)
and e = - 1/2 + sqrt(3)/2*i

Retracing our steps, we see that
1/(3*cbrt(x)^2)*
( e^2*ln(1 - e^2*cbrt(x))
+ e*ln(1 - e*cbrt(x))
+ ln(1 - cbrt(*x)) )

is the value of S, the sum
1/2 + x/5 + x^2/8 + ...
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