Prove by induction on n that, for n a positive integer, (1)/((1-z)^n)=∑k=0^∞( n+k-1 k

step 1 So we're dealing with proofs via induction. We have two theorems to prove here. Theorem A is the

Prove by induction on n that, for n a positive integer,...

Key Concepts

Radius of Convergence

The radius of convergence defines the interval within which a power series converges to the function it represents. Understanding this concept ensures that the series expansion is valid and correctly represents the function for values within the specified range.

Power Series

A power series is an infinite sum of the form sum a?z? that represents a function within a certain interval. It is a fundamental concept in analysis, providing a way to express complex functions as an infinite series of polynomial terms, which is particularly useful when analyzing functions and solving differential equations.

Generalized Binomial Theorem

The generalized binomial theorem extends the classic binomial theorem to cases where the exponent is not necessarily a non-negative integer, allowing expressions like (1 - z)^(-n) to be written as an infinite series. This theorem is instrumental in connecting combinatorial identities with analytic series representations.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a proposition holds for all positive integers by first proving it for a base case and then showing that if it holds for an arbitrary integer, it must hold for the next one. This method is particularly effective for proving properties that are defined recursively or sequentially.

Binomial Coefficients

Binomial coefficients, typically denoted as combinations, quantify the number of ways to choose a subset from a larger set and appear as coefficients in the expansion of binomial expressions. They are crucial in expressing series expansions and play a central role in combinatorial arguments.

Transcript

00:01 So we're dealing with proofs via induction.

00:03 We have two theorems to prove here.

00:07 Theorem a is the sum of the first n in natural numbers.

00:11 And theorem b is the sum of the first in natural cubes.

00:15 Now, to do proof by induction, the first step is to try to prove if your theorem is true for some really simple, trivial case.

00:25 For example, if we let r in the first equation be 1, and we test it in our equation here, we get 1 half times 1 plus 1, 2 plus 1⁄2 times 2 or 1.

00:46 So we see that's true.

00:48 So we've done step 1 for equation a.

00:53 Next, you just assume that that theorem holds true for some larger number, k.

01:03 And the hard part is proving that it's true for that larger number by testing it on the next largest number of k plus one.

01:14 And it's kind of confusing, so it's better if i just show you what i mean.

01:19 I'm looking for r plus 1 2, k plus 1 of 3.

01:25 The first r natural numbers.

01:28 I know that must equal 1 to k.

01:34 The first natural numbers plus k plus 1.

01:41 Next we just substitute the value in for the summation symbol.

01:47 That's going to give us 1 half k, k plus 1 plus k plus 1.

01:56 Now it's all algebra.