# Evaluating Finite Sum: Homework Statement

• gruba
In summary: The sum \sum_{k =0}^n is the sum of k = 0, k= 1, k = 2, ..., k = n. In the second sum, the right side, k is increased by 1 but n is still the same (n -1 + 1 = n). In the third, k is increased by 2 but n is still the same (n - 2 + 2 = n).In summary, the problem involves using the binomial coefficient identity and manipulating the given expression to get rid of the k squared term. However, there seems to be a misunderstanding about the notation and the value of n in the summation. Proper understanding of the notation and
gruba

## Homework Statement

Find $\sum\limits_{k=0}^{n}k^2{n\choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k}$

## Homework Equations

-Binomial theorem

## The Attempt at a Solution

I am using the binomial coefficient identity ${n\choose k}=\frac{n}{k}{{n-1}\choose {k-1}}$:

$$\sum\limits_{k=0}^{n}k^2{n\choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k}=\sum\limits_{k=1}^{n-1}\frac{n}{k}{{n-1}\choose {k-1}}(k-1)^2(\frac{1}{3})^{k-1}(\frac{2}{3})^{n-k+2}$$

What am I doing wrong here (sums are not equal)?

How do you compute the exponent of 2/3 ?

jk22 said:
How do you compute the exponent of 2/3 ?

What do you mean? In the sum, $k$ is increased by $1$ and $n$ is decreased by $1$, so in the function $k$ is decreased and $n$ is increased.

I think first we shall note the domain of validity of the binomial formula.

This imply we shall write a term outside the sum.

Are the k modified in the sum out side the binomial formula ?

I suppose the aim of using that formula two times is to get rid of the k squared.

In general, you have ##(x+y)^n = \sum_{k = 0}^n \binom{n}{k} x^k y^{n-k} ##

Take the derivative with respect to ##x## on each sign of the equation and multiply by ##x##.
You get ##nx(x+y)^{n-1} = \sum_{k = 0}^n k \binom{n}{k} x^k y^{n-k} ##.

Now do it again and set ##x## and ##y## to one third and two thirds.

gruba said:
What do you mean? In the sum, $k$ is increased by $1$ and $n$ is decreased by $1$, so in the function $k$ is decreased and $n$ is increased.
Apparently you are trying to do this problem without knowing what $\sum_{k= 0}^n$ means.
n does NOT "decreas", n is fixed, the maximum value of k.

## 1. What is a finite sum?

A finite sum is a mathematical expression that adds a finite number of terms together. It is denoted by the symbol ∑ (sigma) and is commonly used to represent the sum of a series of numbers or variables.

## 2. How do you evaluate a finite sum?

To evaluate a finite sum, you need to follow specific steps. First, identify the pattern of the terms in the sum. Then, use the appropriate formula to find the sum, such as the arithmetic or geometric series formula. Finally, substitute the values of the terms into the formula and simplify the expression to find the sum.

## 3. What is the difference between a finite sum and an infinite sum?

A finite sum has a fixed number of terms, whereas an infinite sum has an endless number of terms. An infinite sum is also known as a series, and it can either converge (approach a finite value) or diverge (go to infinity).

## 4. What are some real-life applications of finite sums?

Finite sums have various applications in different fields, such as finance, physics, and computer science. For example, in finance, finite sums can be used to calculate compound interest on a loan or investment. In physics, they can be used to calculate the displacement of an object with a changing velocity. In computer science, they can be used for data encryption and compression algorithms.

## 5. Can a finite sum have a negative number of terms?

No, a finite sum cannot have a negative number of terms. The number of terms in a finite sum must be a positive integer, as it represents the number of times the terms are added together. If the number of terms is negative, the sum would not have a well-defined value.

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