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What is this?

Posted on 2008-07-23 05:54 By Sandor Szabo ( Maple Rating 2

102)

Categories:

How do I ...? with Maple

I want to obtain an asymptotic formula (n->infinity) for

sum(2^k/k, k=1..n).

Direct method does not work. Probably it is needed some transformation (maybe theory) before it. Could you help me?

Sandor

Hmm i'm not sure if this is

Comment on 2008-07-23 08:38 from Sandor Szabo ( Maple Rating 2

102)

Hmm i'm not sure if this is allowed: take the sequence and replace it by a continuous function

restart:

f:=2^x/x;

int(f,x);

asympt(%,x,2);

convert(%,polynom);

on a sidenote, strange that Maple considers the last term as a polynom, but it works to remove the O-term.

quadrature error

Comment on 2008-07-23 12:36 from Sandor Szabo ( Maple Rating 2

102)

It is usual to approximate a sum by integral, but what can you say about the error?

A similar case

Comment on 2008-07-23 12:49 from alec ( Maple Rating 4

1443)

Look at a similar case, f(x)=2^x. Then analogous calculations would give asymptotic 2^x/ln(2) while actual asymptotic is 2^(x+1). That gives a conjecture that in this case the asymptotic is 2^(n+1)/n.

Alec

take it with a grain of

Comment on 2008-07-23 12:46 from Thomas Unger (

50)

take it with a grain of salt Sandor, it is just an idea.

i would try to ...

Comment on 2008-07-23 13:41 from Axel Vogt ( Maple Rating 4

869)

i would try to evaluate, which gives a LerchPhi and to convert to hypergeometrics, which is a 2F1 and there are papers by Nico Temme for large parameters - may be they cover it, otherwise i would look into his references for other roads

Asymptotic

Comment on 2008-07-23 14:13 from alec ( Maple Rating 4

1443)

It is easy to prove by induction that S_n > 2^(n+1)/(n-1) for n>4. Also, it is easy to prove by induction that if S_m < 2^(m+1)/(m-1-eps) for some (large enough, depending on eps) m, with eps>0, then S_n < 2^(n+1)/(n-1-eps) for all n>m. That proves asymptotic 2^(n+1)/(n-1) for S_n (actually, even more than that, asymptotic would follow from any particular value of eps, say eps=1.)

Alec

Asymptotic in Maple

Comment on 2008-07-23 16:07 from alec ( Maple Rating 4

1443)

Now, I did that by hand. But this also could be found in Maple using equivalent function from algolib library. First, one has to download algolib from INRIA, then add its location at the beginning of libname, i.e. do

libname:= "/path/to/algolib", libname;

and then, since S_n is the coefficient at x^n in the Taylor series expansion of log(1-2*x)/(x-1) at x=0, do

equivalent(log(1-2*x)/(x-1),x,n);
                         (-ln(2))            (-ln(2))
                2 exp(-n)             exp(-n)
                ----------------- + O(---------------)
                        n                    2
                                            n

simplify(%,symbolic);

                                        n
                          (1 + n)      2
                         2        + O(----) n
                                        2
                                       n
                         --------------------
                                  n

Alec

you made my day

Comment on 2008-07-23 16:21 from Axel Vogt ( Maple Rating 4

869)

Alec,

very nice!

Though I always had problems to make the lib run at my PC ...

classic

Comment on 2008-07-23 16:34 from alec ( Maple Rating 4

1443)

I've just put maple.hdb, maple.ind, and maple.lib in classic folder in lib. Restarted Maple after that (i.e. shut it down and then started again), and did

libname:="C:\\Program Files\\Maple 12\\lib\\classic",libname;

and everything worked, even ?algolib (in Classic.)

Alec

Taylor series expansion of

Comment on 2008-07-24 05:23 from Sandor Szabo ( Maple Rating 2

102)

It is very nice!

Here S_n=sum(2^k/k, k=1..n).

I tried to calculate the Taylor series of log(1-2*x) / (x-1) using

convert( log(1-2*x) / (x-1) , FormalPowerSeries)

but Maple's answer is log(1-2*x) / (x-1) .

However S_n involves LerchPhi a 2F1 hypergeometric function, so I think, in principle, the FormalPowerSeries method should give the desired answer.

Sandor

FormalPowerSeries

Comment on 2008-07-24 13:25 from alec ( Maple Rating 4

1443)

Converting to FormalPowerSeries is relatively new to Maple, and works only in a limited number of cases. It would, certainly, be very useful if it worked for much larger class of functions.

The series -log(1-2*x) has coefficient 2^k/k at x^k, and dividing by (1-x) is the same as multiplication by (1+x+x^2+x^3+...) which makes the coefficient at x^n being equal to the sum of coefficients at 1, x, x^2,..., x^n in the original series.

Alec

using guessgf

Comment on 2008-07-24 16:50 from jakubi ( Maple Rating 4

670)

After adding algolib to 'libname':

Sn:=sum(2^k/k,  k=1..n);

                                             n                        n
                                   Pi n I + 2  LerchPhi(2, 1, n) n - 2
                           Sn := - ------------------------------------
                                                    n



seq(simplify(Sn),n=1..10);

                                         256  416  4832  8192  42496  74752
                       2, 4, 20/3, 32/3, ---, ---, ----, ----, -----, -----
                                         15   15   105   105    315    315


with(gfun):
guessgf([0,seq(simplify(Sn),n=1..10)],x);

                                    ln(-1 + 2 x) - Pi I
                                    [-------------------, ogf]
                                            x - 1

equivalent(%[1],x,n):
map(simplify,%) assuming n>0;

                                         (n + 1)       n
                                        2             2
                                        -------- + O(----)
                                           n           2
                                                      n

}

asymptotic

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