## Writing Series in maple...

f=sum((2*q*cos(2* i*x)*(-1)^(i)*(-1)^((2*i-1)))/(i*Pi),i=1.3.5...35)

I want to write this series but getting error

the result is

2*q*cos(2*x)/Pi-2*cos(6*x)*q/(3*Pi)+2*q*cos(10*x)/(5*Pi)-2*q*cos(14*x)/(7*Pi)+2*q*cos(18*x)/(9*Pi)-2*q*cos(22*x)/(11*Pi)+2*q*cos(26*x)/(13*Pi)-2*q*cos(30*x)/(15*Pi)+2*q*cos(34*x)/(17*Pi)-2*q*cos(38*x)/(19*Pi)+2*q*cos(42*x)/(21*Pi)-2*q*cos(46*x)/(23*Pi)+2*q*cos(50*x)/(25*Pi)-2*q*cos(54*x)/(27*Pi)+2*q*cos(58*x)/(29*Pi)-2*q*cos(62*x)/(31*Pi)+2*q*cos(66*x)/(33*Pi)-2*q*cos(70*x)/(35*Pi)

can anybody help

## Taylor series of an abstract function...

Hey all,

my goal is to get a taylor expansion in Maple up to n-th order of a function y(t) (not defined in more detail, just abstract).

I want to taylor this function around tn, and I want the series for tn+1=tn + h. I would expect to get (as an example: 4th order)

y(tn+1) ≈ y(tn) + h* y'(tn) + h2/2*y''(tn) + h3/6*y'''(tn) + O(h4).

My command was:

and I get:

Can someone maybe point out what I am doing wrong and how I can fix it?

Tim

## time series troubles...

Dear Community,

I have two measured series - time series actually, stored in Excel:

• gas flow (ordered in ascending date order),
• pressure measurements (unordered, more or less randomly distributed dates),

I would like to:

• import the two series into Maple,
• convert them to time series,
• merge them in a way, that the unordered pressure dates are plugged into the ordered gas flow dates, so I finally get a single, ordered date sequence of ascending order.
• pressure and gas flow rates should remain separated of course, i.e. not merged (they should appear in two separate vectors, or matrix columns, whatever)
• if pressure dates and gas flow dates are the same, they should naturally be in the same row, i.e. belong to the same date.

No problems with the import, but I'm lost with the time series handling. I tried to figure it out from the help, but frankly the description is rather difficult, and I also haven't found appropriate examples. Is it possible to do it with Maple's time series feature at all, or something other is more practical? Files and my initial trial are attached.

best regards

Andras

Time_Series_Data.xlsx

TimeSeriesTest.mw

## how do I plot a solution with series representatio...

Hi! I'm trying to find the way to plot the solution with series representation. I need some help to find the easiest way.

Note: I realized some typing errors, which do not change the question a lot ,and I corrected them.

plot.mw

## Please some help for the better study of number of...

I wish to study the trend of medical consultations each day during six years. Thus I expect near 2200 datas to analyse.

But some parameters are to consider :

- I don't have yet the datas per day, but the mean is about 2 consultations per day

- as it is difficult to do more than 3 or perhaps 4 consultations during one day (9h30 am - 13h pm), the others if they exist will probably be seen the next day (aso if the next day 3 news consultations occured)

- then, I don't know actually (as I expect the datas or each day but don't have now these datas) if the better distribution will be simply follow a Poisson' law, or exponential, or negative binomial, ..

- do someone have a clue for the better law given what i said ?

Further, I don't have a stastic program especially used for time trend, excepting Systran 13, but I don't believe that this program can be used with a theoric model of distribution, I recall that it does usual tasks, autocorrelations, saisonnal adjustments, .. but with continuous distributions I believe, and a linear model (removing the basic frequencies)

As such program (study of temporal series) is usually sold about 3000\$ in France, that I don't expect to be a trader, with only one calculus to do, could anyone tell me how to adjust the better model to the 2200 datas that could be expected ?

Thx for your help, friendly yours;

Milos

## geometric series convergence...

Dear all

I have a simple question about geometric series

What's the difference between g(2) and gg(2)

Many thanks

Geometric_series.mw

## Expanding integral of series ...

Dear all

If its possible in  Maple to change the integral of the sum to  the sum of integrals when I calucle the integral of a function series

Thank you

## Truncation of small terms...

Hello there,

Suppose I have a parametric experssion like   P(ε)=1+ε+ε23 where ε is very small. How can I get P(ε)≈1+ε or P(ε)≈1+ε+ε2

Thanks

## How to find all ideals in this?...

Dear All

Using Lie algebra package in Maple we can easily find nilradical for given abstract algebra, but how we can find all the ideal in lower central series by taking new basis as nilradical itself?

 >
 >
 (1)
 Euc >
 (2)
 Euc >
 (3)
 Euc >
 (4)
 L1 >
 L1 >
 (5)
 L1 >
 (6)
 L1 >
 (7)

Taking N as new basis , how we can find all ideals in lower central series of this solvable ideal N?

Regards

## Sums of certain series and a bug...

This question is related to the recent post
http://www.mapleprimes.com/questions/211460-Series-Of-Bessel-Functions

1. Consider the following fast convergent series:

f:=n->(-1)^(n+1)*1/(n+exp(n));
S1:=Sum(f(n),n=1..infinity);
evalf(S1);
S2:=Sum(f(2*n-1)+f(2*n),n=1..infinity);
evalf(S2);

As expected, the sum of the series is obtained very fast (with any precision), same results for S1 and S2.

2. Now change the series to a very slowly convergent one:

f:=n->(-1)^(n+1)/sqrt(n+sqrt(n));

evalf(S1) is computed also extremely fast, because the acceleration algorithm works here perfectly.
But evalf(S2) demonstrates a bug:

Error, (in evalf/Sum1) invalid input: `evalf/Sum/infinite` expects its 2nd argument, ix, to be of type name, but received ...

3. Let us take another series:

f:=n->(-1)^(n+1)/sqrt(n+sqrt(n)*sin(n));

Now evalf(S1) does not evaluate numerically and evalf(S2) ==> same error.
Note that I do not know whether this series is convergent or not, but the same thing happens for the obviously convergent series

f:=n->(-1)^(n+1)/sqrt(n^(11/5)+n^2*sin(n));

(because it converges slowly (but absolutely) and the acceleration fails).
I would be interested to know a method to approximate (in Maple) the sum of such series.

Edit. Now I know that the mentioned series

converges (but note that Leibniz' test cannot be used).

Before version 2016 Maple was incredibly good at evaluating an infinite power series and returning a simple function, e.g. 1/(3x+2).  Now version 2016 just returns the input sum expression with no change.  Is there some new command to get the old results?

## Problem with power series ...

Hello, I have a somewhat math and Maple question I'm hoping some can help with.

I have this curve,

curve:=alpha^2*beta^10-alpha^2*beta^9+4*alpha*beta^7-2*alpha*beta^6-2*alpha*beta^5-alpha*beta^4+alpha*beta^3-2*beta+1;

and if I solve this system (numerically);

sol:= solve([curve=0,diff(curve,beta)=0],[alpha,beta]);

I get 6 special points (8 actually but two are critical). So I'll refer to them by subscript "i".

If I then do a coordinate transformation by;

alpha=exp(u)

beta=exp(v)

and convert the 6 coordinates and curve, I'll get everything in terms of (u,v) coordinates.

curve2:=subs([alpha=exp(u),beta=exp(v)],curve);

soluv:=map(ln,sol);  (this is just pseudo - I don't know how to do it this way)

So now the 6 points are referred to by (u_i,v_i).

Next, I want to expand this curve locally around these six points, using the following (where "z" is the local coordinate);

u -> u_i + z^2

v -> v_i + sum(a_j*z^j,j=1..n)

where n is reasonable, though around 15.

curve3(i):=subs(u=z^2 + cat(Ubp,i),curve2);

curve4(i,n):=subs(v=cat(Vbp,i)+sum(a[i,j]*z^j,j=1..n),curve3(i));

Here, I'm not sure of the pros/cons of cat() vs a[i,j]....

Anyway, I'll then have an equation in terms of only the local coordinate "z". If I then solve each coefficient of "z", at each order, I should then be able to determine the power series v(z). I reason that since the curve is initially equal to zero, that every non-zero power of z will have a coefficient/equation (in terms of unknowns a[i,j]) that should be equal to zero.

This is analagous to solving differential equations with power series...

However, I'm a little lost in implementing this,

I am currently trying, for instance,

l1:= series(curve4(1,10),z,0,9);

e1:= seq(coeff(l1,z,i),i=1..4):

s1:=seq(a[1,j],j=1..4):

sol:=solve(e1,s1);

I believe I am doing something wrong though bcause every odd power is zero.

Thank you a lot for any suggestions and/or help,

sbh

PS>

Ubp and Vbp are those 6 points - I just kept them as symbols initially because I was still getting odd-powered coefficients as zero, regardless of their actual values. Hence a little bit of the "math" side of the problem...

## Does the principal value of this integral exist?...

I want to make sense of the expression

Int(t^2/ln(t)*exp(-t), t=0..infinity);

The denominator vanishes at t=1.  The singularity at t=1 is not integrable.  I want to see whether the integral is defined in the sense of Cauchy principal value.  Thus, I let

K := Int(t^2/ln(t)*exp(-t), t=0..1-a) + Int(t^2/ln(t)*exp(-t), t=1+a..infinity);

and wish to see whether the following limit exists:

limit(K, a=0, right);

Maple cannot evaluate this.  Nor can I.  Alternatively, we may try:

series(K, a=0);

or

series(K, a=0) assuming a>0, a<1;

In both cases Maple says that it is unable to compute the series.

So my question is: Does the Cauchy principal value exist, and can Maple help one to determine that?

## Period ...

Thanks for helping.

(1) But how to find period using code?

(2) how did you determine n=29 ?

(3) I don't need sequence of numbers, but would like to extract the coefficients of exp(..) terms. For example in the follwoing expression, how to extract the coefficients: (note that the expression is missing one more term and i don't know how to fix?)

## Hankel function, series, error bound...

Dear all;

I open this good discussion, and hope can get a nice and strong idea in this domain of approximation of Hankel funciton and order truncation of infinite series. Thanks for all idea, can improve the discussion.

Using the asymptotics of Hankel function for large argument and large orders ( both together) and
find   an order of truncation N of the obove series so that we can ensure an error bound  of epsilon( epsilon very small given).  abs(sum ( c[m]*HankelH1(m,x)*exp(I*m*theta), m=-infinity..infinity)-sum ( c[m]*HankelH1(m,x)*exp(I*m*theta), m=-N..N))<epsilon.

A first idea come in mind: the series converge, so that the general terms of this series converge to zero, and in particularity,  abs(c[m]* HankelH1(m,x))<1:
then abs(c[m])<1/ abs(HankelH1(m,x)).
So we can ensure an error bound of epsilon on the coefficient c[m] by imposing  (HankelH1(m,x))<epsilon  this lead to abs(c[m])<epsilon.
I consider the case where m and x are very large, I can suppose for example m=x*(1+zeta), with 0<zeta<1. zeta parameter. So that our truncation N depend on zeta.
And then how can I find and approximation of the inverse of Hankel function for large argument and large order. using m=x*(1+zeta). I think this give us N the truncation order.
I hope get a good discussion in this subjet.
Of course maybe there are other strong idea to compute the truncation series.
I get the following error in the code:

Error, (in MultiSeries:-multiseries) unable to expand with respect to parameter

###### Code### and error
restart:

with(MultiSeries):
assume(0 <= x);

assume(0 <= zeta<1);

HankelH1(v,x):

sum(c[m]*HankelH1(m, x)*exp(I*m*theta),m=-infinity..infinity);

1/HankelH1(x*(1+zeta),x);

MultiSeries:-asympt(%,x, 4);

eval(%, O=0);

convert(%,exp);

simplify(%);

truncated_series.mw

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