# Help on prooving a summation formula

• nrgyyy
In summary, the conversation is about proving a formula involving an arbitrary function and two summations. The first summation is from t=1 to N and the second summation is from s=1 to N, while the second equation involves a summation from τ=-N+1 to N-1. The person asking for help has tried small examples and is considering changing the variables in the first double summation, but is stuck. The other person suggests considering τ=t-s as a change of index and asks for clarification on the limits of the summations in that case.

## Homework Statement

I need some advice on prooving this formula (f is an arbitrary function):

$\sum^{N}_{t=1}$$\sum^{N}_{s=1}$f(t-s)=$\sum^{N-1}_{τ=-Ν+1}$(N-|τ|)f(τ)

What have you done so far? Have you tried small examples, like N = 2 or N = 3?

RGV

Well its easy to see that it works with examples like N=2 or N=3. For example for N=2 the value of both sides is f(1)+f(-1)+2f(0). Same for N=3. I am thinking that maybe I should do a variables change in the first double sums, to end up to a more common summation formula, but I am kinda stuck.

EDIT: We can consider that s and t are integers, or that the arbitrary f() function represents a discrete time signal.

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Looks to me like there is a change of index at work there. Since the left side of the equation involves f(t- s) and the other side f(τ), you should immediately think of τ= t- s.

could you please help on how the limits of the left side sums would be in that case (τ=t-s)?

## 1. What is a summation formula?

A summation formula is a mathematical expression that allows you to calculate the sum of a series of numbers. It is denoted by a capital Greek letter sigma (Σ) and is followed by the numbers to be summed, along with their corresponding coefficients. For example, the summation formula for the series 1+2+3+4+5 would be written as Σn, where n represents the number being summed.

## 2. How do I prove a summation formula?

To prove a summation formula, you will need to use mathematical induction. This method involves showing that the formula holds true for a specific value (usually 1 or 0), and then proving that if the formula holds for a particular value, it also holds for the next value. By repeating this process, you can show that the formula holds true for all values in the series.

## 3. What are the steps to proving a summation formula using mathematical induction?

The steps to proving a summation formula using mathematical induction are as follows:

Step 1: Show that the formula holds true for the first value in the series (usually 1 or 0).

Step 2: Assume that the formula holds true for a particular value (usually denoted as k).

Step 3: Use this assumption to prove that the formula also holds true for the next value (k+1).

Step 4: Repeat this process for all values in the series, showing that the formula holds true for each value.

Step 5: Conclude that the formula holds true for all values in the series, thereby proving its validity.

## 4. Can a summation formula be used for infinite series?

Yes, a summation formula can be used for infinite series. However, in order for the formula to be valid, the series must be convergent, meaning that the sum of the series approaches a finite value as the number of terms increases. In these cases, the summation formula is denoted by the infinity symbol (∞), and the series is written as Σn=1 to ∞.

## 5. What are some common summation formulas?

Some common summation formulas include:

- Sum of integers: Σn = n(n+1)/2

- Sum of squares: Σn^2 = n(n+1)(2n+1)/6

- Sum of cubes: Σn^3 = (n(n+1)/2)^2

- Geometric series: Σar^n = a(1-r^n)/(1-r), where a is the first term and r is the common ratio

- Harmonic series: Σ1/n = ln(n+1), where ln is the natural logarithm