Homework Help: Verifying summation with Maple

1. The problem statement, all variables and given/known data

Verify the following summations using Maple (see image).

2. Relevant equations

None

3. The attempt at a solution

For the first one, I enter

Code (Text):

sum(k^3, k=1..n);

in Maple, and the result is

1/4*(n+1)^4-1/2*(n+1)^3+1/4*(n+1)^2,

which is definitely not the answer. Why is Maple giving me these weird and incorrect answers? The question is rather simple as I am just supposed to verify these summations using a computer, but for some reason, my computer seems to be complicating things.

 

Okay, so it seems if I type

Code (Text):

factor(sum(k^3, k=1..n));

,

I get the correct answer. What gives?

 

Similarly,

Code (Text):

sum(x^k, k=0..infinity);

1
- -----
x - 1

which = 1/(1-x).

But when I type

Code (Text):

evalb(-1/(x-1) = 1/(1-x));
                               

 

I obtain false. But they are equal! Am I completely missing something with Maple here???

 

Oh, I understand that completely. I think most people on this forum do. What is puzzling me is that when I am given the expanded form, I don't immediately recognize it as being the correct answer because I am not quite a mathematician yet. So when I try to see if the expanded expression is equal to the factored expression, by using the evalb() operator in Maple, I get "false". I think one of my posts explained that. That's what's puzzling.

In other words, Maple is telling me 1/(1-x) =/= -1/(x-1), but they are, in fact, equal. This is my issue.