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 $$\big (1+2+\ldots+k+ (k+1)\big)^2- (1+2+\ldots+k)^2\;.$$ …

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 …

Dec 9, 2014 · 65 $\begingroup$ This question already has answers here: Proving the identity $\sum_ {k=1}^n {k^3} = \big (\sum_ {k=1}^n k\big)^2$ without induction (31 answers)

Nov 21, 2018 · i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?

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...

Mar 25, 2013 · Need guidance on this proof by mathematical induction. I am new to this type of math and don't know how exactly to get started. $$ 1^3 + 2^3 + 3^3 + \ldots + n^3 = \left [\frac {n (n+1)} {2}\

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 ...

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 ...

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_ …

$$\\sum_{n=1}^\\infty \\frac{n}{3^n}$$ How do you find the sum? I don't know how to start this problem and no other website I found talks about a problem like this.