In Exercises 41-47, we define the first difference δf of a function f by δf(x)=f(x+1)-f(x). Supp

step 1 In this problem we have to define the first difference delta f of function f by delta f of x is

In Exercises 41-47, we define the first difference δf of a...

In Exercises $41-47,$ we define the first difference $\delta f$ of a function $f$ by $\delta f(x)=f(x+1)-f(x)$. Suppose we can find a function $P$ such that $\delta P(x)=(x+1)^{k}$ and $P(0)=0 .$ Prove that $P(1)=1^{k}, P(2)=1^{k}+2^{k},$ and, more generally, for every whole number $n$ $$P(n)=1^{k}+2^{k}+\cdots+n^{k}$$

In Exercises $41-47,$ we define the first difference $\delta f$ of a function $f$ by $\delta f(x)=f(x+1)-f(x)$.
Suppose we can find a function $P$ such that $\delta P(x)=(x+1)^{k}$ and $P(0)=0 .$ Prove that $P(1)=1^{k}, P(2)=1^{k}+2^{k},$ and, more generally, for every whole number $n$
$$P(n)=1^{k}+2^{k}+\cdots+n^{k}$$

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Key Concepts

First Difference Operator

The first difference operator is a discrete analogue to the derivative in calculus. Instead of measuring the instantaneous rate of change, it measures the change in a function’s values between consecutive integer points. This concept is fundamental in studying discrete functions and sequences, as well as in developing difference equations which serve a role similar to differential equations in continuous contexts.

Telescoping Sum

A telescoping sum is a series in which many terms cancel out when the series is expanded, leaving only a few terms that determine the overall sum. This property is commonly used in discrete mathematics and summation proofs, as it allows one to simplify the sum of differences or any series where consecutive terms negate some parts of one another.

Discrete Summation

Discrete summation refers to the process of summing the values of a function over a set of discrete points, typically integers. In the context of the problem provided, the function P is constructed such that its first difference reproduces a power function; consequently, by summing these differences, one recovers the cumulative sum of powers over an interval. This technique is analogous to the process of integration in continuous calculus and is a cornerstone in finite calculus when dealing with series and summations.

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