# Simplify a series expansion including product and multiplication

I have a following expression $$f=\Pi_{i=1}^n \left[ 1 + \frac{1}{t} +\frac{ (m+i)}{t^2} +\frac{ (m+i)^2}{t^3} + \cdots \right]$$ here $m,n,t$ are positive integers.

I want to obtain a series expansion for $f$ as $1/t,1/t^2,1/t^3,\cdots$ up to maybe $1/t^6$. A general expression for any order is not needed.

I used

f = \!$$\*UnderoverscriptBox[\(\[Product]$$, $$i = 1$$, $$n$$]$$(1 + \*FractionBox[\(1$$, $$t$$] +
\*FractionBox[$$(m + i)$$, $$t^2$$] +
\*FractionBox[$$\((m + i)$$^2\), $$t^3$$])\)\)

Series[f, {t, Infinity, 3}, Assumptions -> Element[m, Integers]]


Mathematica 7.0 gives me a complicated expression including Arg, Floor, Csc, etc.

Is there any way to obtain simple expression? I can do low order by hand, but for high order, it is increasingly complicated..

• Welcome to Mathematica.SE! Could you please add the actual code you evaluated before the Series function? % just refers to the prior evaluation, and we don't know exactly what you've done. Oct 16, 2014 at 4:11
• Thanks. It is just the expression for f. f = \!( *UnderoverscriptBox[([Product]), (i = 1), (n)]((1 + *FractionBox[(1), (t)] + *FractionBox[((m + i)), (t^2)] + *FractionBox[(((m + i))^2), (t^3)]))) Oct 16, 2014 at 4:12
• I have replaced % as f. In the input, only the expression up to $t^{-3}$ inside multiplication was used. Oct 16, 2014 at 4:14
• I have added f, copied from mathematica. I do not know what is the best way to present it on stackexchange though... Oct 16, 2014 at 4:18

EDIT: Added verification of Product

f can be expressed in closed form:

f[m_, n_, t_] = Product[1 +
Sum[(m + i)^k/t^(k + 1), {k, 0, Infinity}],
{i, 1, n}]


(m - t)/(m + n - t)

Verifying the Product (proof by induction)

s[i_] = 1 + Sum[(m + i)^k/t^(k + 1), {k, 0, Infinity}]


1 - 1/(i + m - t)

f[m, 1, t] == s[1] &&
f[m, n + 1, t] == f[m, n, t]*s[n + 1] //
Simplify


True

Your desired expansion is

Series[f[m, n, t], {t, Infinity, 4}] // Simplify


The coefficients are

SeriesCoefficient[f[m, n, t], {t, Infinity, k}]


Verifying the expansion

f[m, n, t] == 1 + Sum[n (m + n)^(k - 1)/t^k, {k, 1, Infinity}] // Simplify


True

• Thanks for your answer. Excuse me, I do not know how the closed form expression was derived. When I sum the infinity series, I got $$1 + \frac{1}{t-m-i}$$. How to show $$\Pi_{i=1}^n 1 + \frac{1}{t-m-i} = \frac{ m-t}{m+n-t}$$? Oct 16, 2014 at 5:30
• Evaluating Product[1 + 1/(t - m - i), {i, 1, n}] in Mma gives (m-t)/(m+n-t) Oct 16, 2014 at 5:41
• I mean the details of derivation, not how mathematica gives the answer... Oct 16, 2014 at 5:50
• Got it, just write the formula explicitly, $\Pi_{i=1}^n 1 + \frac{1}{t-m-i} = \frac{ t-m}{t-m-1} \frac{t-m-1}{t-m-2} \frac{t-m-2}{t-m-3} \cdots \frac{t-m-(n-1)}{t-m-n}$ and cancel the same terms on denominator and numeriator Oct 16, 2014 at 6:34

Assuming your product is one that Mathematica doesn't know the closed form for, you could use my answer from Infinite product with O[x] here:

Unprotect[Series];

Series[Product[a_, b_], pt_] := Exp[MapAt[Sum[#, b]&, Series[Log[a], pt], 3]]
Series[Inactive[Product][a_, b_], pt_] := Series[Unevaluated[Product[a, b]], pt]

Protect[Series];


Then:

f = Inactive[Product][1 + 1/t + (m + i)/t^2 + (m + i)^2/t^3, {i, 1, n}];

Simplify @ Series[f, {t, Infinity, 3}] //TeXForm


$1+\frac{n}{t}+\frac{n (m+n)}{t^2}+\frac{n (m+n)^2}{t^3}+O\left(\left(\frac{1}{t}\right)^4\right)$