# Infinite summation with absolute value

Recently I need to compute the summation like the following form:

Sum[t^Abs[n - m], {n, -∞, ∞}, Assumptions -> m ∈ Integers]


But Mathematica cannot figure it out:

Although it would be simple to compute without m:

I wonder why and how could I solve it?

• Note that the second result is not correct without additional assumptions. Jul 11, 2016 at 21:19

This sum is indeed possible to compute as it is using the FourierSequenceTransform. Some algorithms of MA are obviously more advanced than others:

s = FourierSequenceTransform[t^Abs[n - m], n, w, Assumptions -> m \[Element] Integers]
(*(E^(-I (-1 + m) w) (-1 + t^2))/((E^(I w) - t) (-1 + E^(I w) t))*)


Now we recall that for $w\rightarrow 0$ the desired sum is obtained

Cancel[s/. w -> 0]
(*(1 + t)/(1 - t)*)


And, as @Michael E2 noticed, one should pay attention to the radius of convergence of the sum.

• Thank you! But how about a even complex summation, like this one: Sum[t^(1/2 (Abs[n - m1] + Abs[n - m2]) + 3/2 (Abs[m1] + Abs[m2]) + m2 - m1), {n, -\[Infinity], \[Infinity]}, {m2, -\[Infinity], \ \[Infinity]}, {m1, m2, \[Infinity]}] This summation should have a simple result, as I have computed it manually. But mathematica can't solve this summation using above method Jul 11, 2016 at 22:44
• @user27490 How have you reformulated your sum m1to run in infinite limits? Only then it will be suitable for Fourier methods. Jul 12, 2016 at 13:47

It seems to help to explicitly expand Abs, which by default is a function over the complexes, for a real/integer argument:

Assuming[
{k, m} ∈ Integers,
Sum[t^Abs[k - m] // PiecewiseExpand, {k, -∞, ∞},
Assumptions -> m ∈ Integers && Abs[t] < 1]
]
(*  (-1 - t)/(-1 + t)  *)


Follow-up to comment:

Maybe deconstruct the pieces semi-manually? (I assume Sum does something like this above, but gives up here.)

summand = Assuming[{k, m1, m2} ∈ Integers,
t^(1/2 (Abs[k - m1] + Abs[k - m2]) + 3/2 (Abs[m1] + Abs[m2]) + m2 - m1) //
PiecewiseExpand[#, Method -> {"ConditionSimplifier" -> Simplify}] &
];
Simplify@ Total@ Assuming[{k, m1, m2} ∈ Integers,
Sum[Piecewise[{#}, 0],
{k, -∞, ∞}, {m2, -∞, ∞}, {m1, m2, ∞},
Assumptions -> k ∈ Integers && m1 ∈ Integers && m2 ∈ Integers && Abs[t] < 1] & /@
Append[First@summand, {Last@summand, Or @@ summand[[1, All, 2]] // Not // Reduce}]
]
(*  -((1 + 4 t + 4 t^2 + 5 t^3 + 4 t^4) / ((-1 + t)^3 (1 + t + t^2)))  *)

• Thank you! But how about a even complex summation, like this one: Assuming[{n, m1, m2} \[Element] Integers, Sum[t^(1/2 (Abs[n - m1] + Abs[n - m2]) + 3/2 (Abs[m1] + Abs[m2]) + m2 - m1) // PiecewiseExpand, {n, -\[Infinity], \[Infinity]}, {m2, \ -\[Infinity], \[Infinity]}, {m1, m2, \[Infinity]}, Assumptions -> n \[Element] Integers && m1 \[Element] Integers && m2 \[Element] Integers && Abs[t] < 1]] This summation should have a simple result, as I have computed it manually. But mathematica can't solve this summation using above method. Jul 11, 2016 at 22:42
• @user27490 : Instead of adding essentially the same comment to every answer, could you instead update your question so that it is actually the question you want answered? Jul 12, 2016 at 5:36