I think this would constitute at least a medium size project if you were to do all that you might want. You can probably do a lot of things with the built-in SeriesCoefficient, SeriesData and ComposeSeries.
In any case, here is a start. In order to display manipulations of the series we need a function that mimics an infinite series without attempting evaluation. We would only evaluate when we wanted to - if ever. To follow this you should look at the Help for TemplateBox and MakeBoxes.
We define such an infiniteSum expression and MakeBoxes for it so it will display as a usual infinite sum. The we define a rule for summing two series and a routine to convert an infiniteSeries into a finite sum with a given number of terms. To get the DisplayFunction Boxes I typed the generic Sum expression into the notebook, evaluated it, and then used Show Expression from the Cell Menu to see the Box structure for the Mathematica expression. I then copied that into the TemplateBox code.
infiniteSum::usage = "infiniteSum[{n,t},term] represents the infinite sum of term[n]t^n";
infiniteSum /:
MakeBoxes[infiniteSum[{n_, t_}, u_],
form : (StandardForm | TraditionalForm) : StandardForm] :=
TemplateBox[{MakeBoxes[n, form], MakeBoxes[t, form],
MakeBoxes[u, form]}, "infiniteSum",
DisplayFunction :> (RowBox[{UnderoverscriptBox["\[Sum]",
RowBox[{#1, "=", "0"}], "\[Infinity]"],
RowBox[{SuperscriptBox[#2, #1], " ", #3}]}] &),
InterpretationFunction :> (RowBox[{"infiniteSum", "[", "{", #1,
",", #2, "}", "," #3, "]"}] &),
Editable -> True,
Selectable -> True]
sumRule =
infiniteSum[{n_, t_}, a_] + infiniteSum[{n_, t_}, b_] :>
infiniteSum[{n, t}, a + b];
infiniteSumSeries[terms_][expr_] :=
expr /. infiniteSum[{n_, t_}, u_] :>
Sum[Evaluate[u t^n] /. n -> i, {i, 0, terms}]
Then as a demonstration let's do a Sin series, Cos series and their sum. I'm a little too lazy to copy all the formatted output into this posting, but if you copy to a notebook and evaluate everything should look good.
sinC = SeriesCoefficient[Sin[t], {t, 0, n}, Assumptions -> n >= 0];
cosC = SeriesCoefficient[Cos[t], {t, 0, n}, Assumptions -> n >= 0];
Series[Sin[t] + Cos[t], {t, 0, 10}]
The following starts with the sum of two generic series, combines them with the sumRule, substitutes the Sin and Cos coefficients and simplifies. Then we generate 10 terms of the series and see that they match the expected expression.
step1 = infiniteSum[{n, t}, u[n]] + infiniteSum[{n, t}, v[n]]
Print["Combine with the sumRule"]
step2 = step1 /. sumRule
Print["Substituting expressions for the Sin and Cos coefficients and \
Simplifying"]
step3 = step2 /. u -> Function[n, sinC] /. v -> Function[n, cosC]
step4 = FullSimplify[step3, n \[Element] Integers && n >= 0]
step4 // infiniteSumSeries[10]
That is just a start because there are sure to be a number of other operations you would need.