summation - Prove that $1^3 + 2^3 + ... + n^3 = (1+ 2 + ... + n)^2 ...
HINT: You want that last expression to turn out to be $\big (1+2+\ldots+k+ (k+1)\big)^2$, so you want $ (k+1)^3$ to be equal to the difference $$\big (1+2+\ldots+k+ (k+1)\big)^2- (1+2+\ldots+k)^2\;.$$ …
Proving $1^3+ 2^3 + \cdots + n^3 = \left (\frac {n (n+1)} {2}\right)^2 ...
Dec 9, 2014 · Hint $ $ First trivially inductively prove the Fundamental Theorem of Difference Calculus $$\rm\ F (n) = \sum_ {k\, =\, 1}^n f (k)\, \iff\, F (n) - F (n\!-\!1)\, =\, f (n),\ \ \, F (0) = 0\qquad$$ The …
Big-O Notation - Prove that $n^2 - Mathematics Stack Exchange
Jul 5, 2013 · Would this book be Kenneth H. Rosen's "Discrete Mathematics and its Applications" by any chance? If so, I found all of the material easily digestible until this section, where I feel they really …
how to prove that $f (n)=n^3+n\log^2n$ = $\theta (n^3)$?
Nov 21, 2018 · i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?
Use mathematical induction to prove that $n^ 3 − n$ is divisible by 3 ...
Solution: Let $P(n)$ be the proposition “$n^3−n$ is divisible by $3$ whenever $n$ is a positive integer”. Basis Step:The statement $P(1)$ is true because $1^3 ...
Proof that $n^3+2n$ is divisible by $3$ - Mathematics Stack Exchange
Let n^3+2n = P (n). We know that P (0) is divisible by 3. The inductive step shows that P (n+1) = P (n) + (something divisible by 3). So if P (0) is divisible by 3, then P (1) is divisible by 3, and then...
Series convergence test, $\sum_ {n=1}^ {\infty} \frac { (x-2)^n} {n3^n}$
Jun 28, 2020 · By the ratio test, every x value between -1 and 5 would make the series converge. we just need to find out whether x=-1, 5 makes it converge. x=-1: The series will look like this. $$\sum_ …
algorithms - How to arrange functions in increasing order of growth ...
Given the following functions i need to arrange them in increasing order of growth a) $2^ {2^n}$ b) $2^ {n^2}$ c) $n^2 \log n$ d) $n$ e) $n^ {2^n}$ My first attempt ...
How to prove that $6n^2 + 20n \notin \Omega { (n^3)}$
Sep 1, 2016 · I'm not sure how to go about this, I've tried finding constants and N values to find out, for instance I tried to prove it through contradiction but I must be doing something wrong because I'm gett...
Recursive definition of natural numbers - Mathematics Stack Exchange
Jul 11, 2015 · I'm doing the exercises in Algorithms and Data Structures in Java, Second Edition, by Adam Drozdek. One question is: The set of natural numbers $\\mathbb{N}$ defined at the beginning …