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Sorry, I think I made a mistake in my origial code, so I used your 'giac' algorithm and all works now. sage: reset() sa

Thank you, but the right limit, .limit(x=-2,algorithm='giac',dir='+'), returns 2. but it should be -5.

Simple Piecewise limit? Here's the question: $$\lim _{x \rightarrow-2} f\left(x\right) =?$$ where $$f\left(x\right)= 3 x

Thank you! A link to your answer will be shared with my pre-calculus students. YouTube video link: https://youtu.be/9Ols

complex limits Given that $z$ is a complex number of the form $z = a +b i$, where $a, \, b \in \mathbb{R}$, and $\overli

My guide has recently been updated with embedded videos. You may download a copy at: https://www.dropbox.com/s/6zjkacmoo

The answer would look like this: $$(2n+3)(2n+2)(2n+1)(2n)(2n-1) \cdots (2n-n+2)$$ It would have $n+2$ factors. Not very

The answer would look like this: $$(2n+3)(2n+2)(2n+1)(2n)(2n-1) \cdots (2n-n+2)$$ It would have $n+2$ factors.

The answer would look like this: $$(2n+3)(2n+2)(2n+1)(2n)(2n-1) \cdots (2n-n+2)$$ It would have $n+2$ factors

The answer would look like this: $$(2n+3)(2n+2)(2n+1)(2n)(2n-1) \cdots (2n-n+2)$$

I'm trying to write a reciusive function to compute $$S_p=1^p+2^p+3^p+\dots+n^p,$$ where both $n$ and $p$ are positive integers. My Sage code correctly returns $S_1$, but does not work for $p>1$. Any ideas?

I am not a big Sage user, but I am using it as a way to introduce pre-calculus students to CAS. Hopefully, I will be able to share this post with my students, and certainly, give credit to anyone that helps fix this mess.

Here's the code:

sage: reset()
sage: k,n,p=var('k,n,p')
sage: assume(k,'integer')
sage: assume(n,'integer')
sage: assume(p,'integer')
sage: def S(p):
....:     if p == 1:
....:         return n*(n+1)/2
....:     else:
....:         return ((n+1)^(p+1) - n - 1 - sum(binomial(p+1,k)*S(p-k+1),k,2,p))
....: /(p+1)
....:     
sage: S(1)
1/2*(n + 1)*n
sage: S(2)

Thank you, Max Alekseyev. I will be sure to share this with my students!

Thank you very much. I Will certainly share this with my students. However, I'm really trying to derive a recursive fun

Recursive Function Error I'm trying to write a reciusive function to compute $$S_p=1^p+2^p+3^p+\dots+n^p,$$ where both $

I had the same/similar issue when upgrading to 9.2. I just downgraded to 9.1 and all is fine.

I would, but I don't own a new Apple M1 machine.

I wonder if the code base for Sage will be recompiled for Apple's new M1 chip. Big Sur's XCode supposedly makes the task easy peasy. However, I doubt anything is easy when dealing with massively complex software. Although I am happy with my Intel-based MacBook Air, I will eventually need to upgrade to Apple Silicon. Any thoughts?