Strategy or software for representing infinite series in sigma form

Question

I'm wondering if there is a strategy or much preferably software for converting an infinite series from several terms to sigma form such as mathematica or maple ect? mathematica and maple are good at generating power series term by term but not in sigma form.

This is the series to 10 terms:

$$x-\frac{x^2}{r t}+\frac{2 x^3}{3 r^2 t^2}+\left(-\frac{1}{2 r^3 t^3}-\frac{1}{4 r^2 t^3}-\frac{1}{6 r t^3}\right) x^4+\left(\frac{2}{5 r^4 t^4}+\frac{2}{5 r^3 t^4}+\frac{4}{15 r^2 t^4}\right) x^5+\left(-\frac{1}{3 r^5 t^5}-\frac{1}{2 r^4 t^5}-\frac{5}{12 r^3 t^5}-\frac{5}{36 r^2 t^5}-\frac{1}{15 r t^5}\right) x^6+\left(\frac{2}{7 r^6 t^6}+\frac{4}{7 r^5 t^6}+\frac{25}{42 r^4 t^6}+\frac{1}{3 r^3 t^6}+\frac{46}{315 r^2 t^6}\right) x^7+\left(-\frac{1}{4 r^7 t^7}-\frac{5}{8 r^6 t^7}-\frac{19}{24 r^5 t^7}-\frac{19}{32 r^4 t^7}-\frac{19}{60 r^3 t^7}-\frac{7}{80 r^2 t^7}-\frac{1}{28 r t^7}\right) x^8+\left(\frac{2}{9 r^8 t^8}+\frac{2}{3 r^7 t^8}+\frac{1}{r^6 t^8}+\frac{25}{27 r^5 t^8}+\frac{163}{270 r^4 t^8}+\frac{106}{405 r^3 t^8}+\frac{88}{945 r^2 t^8}\right) x^9+\left(-\frac{1}{5 r^9 t^9}-\frac{7}{10 r^8 t^9}-\frac{73}{60 r^7 t^9}-\frac{4}{3 r^6 t^9}-\frac{149}{144 r^5 t^9}-\frac{691}{1200 r^4 t^9}-\frac{18569}{75600 r^3 t^9}-\frac{761}{12600 r^2 t^9}-\frac{1}{45 r t^9}\right) x^{10}$$

The first step in attempting to convert was to separate into odd and even powers of x:

odd,

$$x+\frac{2 x^3}{3 r^2 t^2}+\left(\frac{2}{5 r^4 t^4}+\frac{2}{5 r^3 t^4}+\frac{4}{15 r^2 t^4}\right) x^5+\left(\frac{2}{7 r^6 t^6}+\frac{4}{7 r^5 t^6}+\frac{25}{42 r^4 t^6}+\frac{1}{3 r^3 t^6}+\frac{46}{315 r^2 t^6}\right) x^7+\left(\frac{2}{9 r^8 t^8}+\frac{2}{3 r^7 t^8}+\frac{1}{r^6 t^8}+\frac{25}{27 r^5 t^8}+\frac{163}{270 r^4 t^8}+\frac{106}{405 r^3 t^8}+\frac{88}{945 r^2 t^8}\right) x^9$$

even,

$$-\frac{x^2}{r t}+\left(-\frac{1}{2 r^3 t^3}-\frac{1}{4 r^2 t^3}-\frac{1}{6 r t^3}\right) x^4+\left(-\frac{1}{3 r^5 t^5}-\frac{1}{2 r^4 t^5}-\frac{5}{12 r^3 t^5}-\frac{5}{36 r^2 t^5}-\frac{1}{15 r t^5}\right) x^6+\left(-\frac{1}{4 r^7 t^7}-\frac{5}{8 r^6 t^7}-\frac{19}{24 r^5 t^7}-\frac{19}{32 r^4 t^7}-\frac{19}{60 r^3 t^7}-\frac{7}{80 r^2 t^7}-\frac{1}{28 r t^7}\right) x^8+\left(-\frac{1}{5 r^9 t^9}-\frac{7}{10 r^8 t^9}-\frac{73}{60 r^7 t^9}-\frac{4}{3 r^6 t^9}-\frac{149}{144 r^5 t^9}-\frac{691}{1200 r^4 t^9}-\frac{18569}{75600 r^3 t^9}-\frac{761}{12600 r^2 t^9}-\frac{1}{45 r t^9}\right) x^{10}$$

EDIT added:

For those that cannot understand my question.

This is an example of a power series put into sigma notation:

what I want is a strategy or software for representing the series terms on the R.H.S in sigma form as shown in the middle. And the specific exapmle I'm working on is in the original question.

Answer 1

In Maple, you can try using the gfun package to guess a generating function or a recurrence for the terms, then convert(..., FPS) to get a closed form for the series of the generating function or rsolve to solve the recurrence. For example:

S:= 1-x+4/3*(x^2)-2*x^3+16/5*(x^4)-16/3*(x^5)+64/7*(x^6)-16*x^7+256/9*(x^8)-256/5*(x^9)+1024/11*(x^10); $$ S:= 1-x+\frac{4}{3}\,{x}^{2}-2\,{x}^{3}+{\frac {16\,{x}^{4}}{5}}-\frac{16}3\,{x}^{5}+{ \frac {64\,{x}^{6}}{7}}-16\,{x}^{7}+{\frac {256\,{x}^{8}}{9}}-{\frac { 256\,{x}^{9}}{5}}+{\frac {1024\,{x}^{10}}{11}} $$

Coeffs:= [seq(coeff(S, x, j), j=0..10)];

$$Coeffs:= \left [1,-1,\frac43,-2,{\frac{16}{5}},-\frac{16}3,{\frac{64}{7}},-16,{\frac{256}{9}},- {\frac{256}{5}},{\frac{1024}{11}}\right] $$

g:= gfun:-guessgf(Coeffs,x);

$$\left[{\frac {\ln \left( 1+2\,x \right) }{2x}},{\it ogf}\right]$$

convert(g[1], FPS, x);

$$\sum _{k=0}^{\infty }{\frac { \left( -1 \right) ^{k}{2}^{k}{x}^{k}}{k+ 1}} $$

or:

rec:= gfun:-listtorec(Coeffs, a(n));

$$[ \left\{ \left( -2\,n-4 \right) a \left( n+1 \right) + \left( -n-3 \right) a \left( n+2 \right) ,a \left( 0 \right) =1,a \left( 1 \right) =-1 \right\} ,{\it ogf}] $$

aa:= rsolve(rec[1],a(n));

$$aa:= {\frac { \left( -1 \right) ^{n}{2}^{n}}{n+1}}$$

Sum(aa*x^n, n=0..infinity);

$$\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}{2}^{n}{x}^{n}}{n+ 1}} $$