I’m getting tripped up by sigma notation and I can’t tell if I’m making a silly off-by-one mistake or misunderstanding the whole setup. When I see something like the sum from n = 1 to 4 of (2n + 1), I think I understand that I’m adding a bunch of odd numbers, but I keep second-guessing whether I’m supposed to include the 4 or not. Is the top number always included? I keep reading it as a range, and my brain does that programming thing where ranges sometimes exclude the end, and then I panic.
Also, I’m confused about “shifting” the index. Like, if I have the sum from k = 0 to 3 of k^2, is that the same as the sum from k = 1 to 4 of (k − 1)^2? It feels like it should be the same list of terms, just re-labeled, but I don’t know if I’m allowed to do that without changing the result. Is there a simple rule of thumb for when you can shift the index and how to do it properly?
And then there’s the whole “first n terms” thing. If a sequence is defined as a_n = 3n − 1 but starts at n = 0, and someone asks for the sum of the first 4 terms, do I use n = 0 to 3 or n = 1 to 4? My answers keep being off by one term depending on where I start, and I can’t figure out a clean way to line up the sigma with what “first 4 terms” means.
One more tiny thing: what about constants? For example, the sum from i = 1 to 5 of 3. Am I literally just adding 3 five times, or is there a smarter way I’m supposed to think about that in sigma land?
Could someone explain how to keep these straight? I feel like I see the pattern but then I shift an index or misread an endpoint and the whole thing derails.
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3 Responses
I’m pretty sure the upper limit is included (so sum n=1 to 4 of (2n+1) means 3+5+7+9), you can safely “shift” by substituting variables to keep the same terms (e.g., sum k=0 to 3 of k^2 equals sum n=1 to 4 of (n−1)^2), “first 4 terms” means the first four indices starting from whatever the sequence starts at (so if it starts at 0, use n=0 to 3), and constants just multiply (sum i=1 to 5 of 3 = 5×3). I used to get ambushed by off-by-one gremlins until a tutor made me list the actual terms beside the sigma like a mini checklist, and this helped too: https://www.khanacademy.org/math/ap-calculus-bc/bc-integration-new/bc-sigma-notation/v/sigma-notation-introduction.
Think of Σ from n=a to b as inclusive counting-like Monday to Friday is 5 days-so you plug in n=a, a+1, …, b (that’s b−a+1 terms); you can shift the index by a substitution m=n+c and shifting limits the same way (e.g., Σ_{k=0}^3 k^2 = Σ_{k=1}^4 (k−1)^2), “first 4 terms” means the first four indices where the sequence actually starts (if it starts at 0 use n=0..3; if at 1 use n=1..4), and constants just stack: Σ_{i=1}^5 3 = 3·5 in general Σ_{i=a}^b c = c(b−a+1).
Want to try expanding your examples term by term to see the lists match up?
I might be mixing conventions, but I think the upper limit isn’t included (so ∑ from n=1 to 4 of (2n+1) stops at n=3), shifting the index doesn’t change the bounds (so ∑ from k=0 to 3 of k^2 is not the same as ∑ from k=1 to 4 of (k−1)^2), “first 4 terms” when a sequence starts at n=0 means sum n=1 to 4, and ∑ from i=1 to 5 of 3 would be 3 added four times (12).
Does your textbook maybe use a different starting-index convention, or is there a specific example where my interpretation doesn’t match the posted solution?