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# Partial sums intro

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.1 (EK)
,
LIM‑7.A.2 (EK)

## Video transcript

let's say that you have an infinite series s which is equal to the sum from N equals one let me write that a little bit neater n equals one to infinity of a sub N and this is all a little bit of review we would say well this is the same thing as a sub 1 plus a sub 2 plus a sub 3 and we would just keep going on and on and on forever now what I want to introduce to you is the idea of a partial sum this right over here is an infinite series but we could define a partial sum so if we say S sub 6 this notation says ok if s is an infinite series s sub 6 is the partial sum of the first six terms so in this case this is going to be we're not going to just keep going on forever this is going to be a sub 1 plus a sub 2 plus a sub 3 plus a sub 4 plus a sub 5 + a sub 6 and I could make this a little bit more tangible if you like so let's say that s the infinite series s is equal to the sum from N equals 1 to infinity of 1 over N squared in this case it would be let's see 1 over 1 squared plus 1 over 2 squared plus 1 over 3 squared and we would just keep going on and on and on forever but what would S sub actually me do that same color what would s I said I would change color and I didn't what would S sub 3 be equal to the partial sum of the first three terms and I encourage you to pause the video and try to work through it on your own well it's just going to be it's just going to be the first term 1 plus the second term 1/4 plus the third term 1/9 it's going to be the sum of the first three terms and we could figure that out that's the C if you have a common denominator here it's going to be 36 to be 36 36 + 9 36 + 4 36 so this is going to be 49 over 36 49 49 over 36 so the whole point of this video is just to appreciate this idea of a partial sum and what we'll see is that you can actually Express what a partial sum might be algebraically so for example for examples give ourselves a little bit more real estate here let's say let's go back to just saying we have an infinite series s that is equal to the sum from N equals 1 to infinity of a sub n and let's say we know the partial sum the partial sum S sub n so the sum of the first n terms of this is equal to n squared minus 3 over over N to the third plus 4 so just those are a bit of a reminder of what this is saying S sub n s sub n is the same thing as a sub 1 plus a sub 2 plus you keep going all the way to a sub N and that's going to be equal to this business N squared minus 3 over n to the third plus 4 now given that if someone were to walk up to you on the street and said okay now that you know the notation for a partial sum I have a little question to ask of you if this is if s is the infinite series and I'm writing it in very general terms right over here so s is the infinite series from N equals 1 to infinity of a sub N and the partial sum S sub n is defined this way so someone they tell you these two things and then they say find find what the sum from N equals 1 to 6 of a sub n is and I encourage you to pause the video and try to figure it out well this is just going to be this is going to be a sub 1 plus a sub 2 plus a sub 3 plus a for when I say sub that just means subscript plus a sub 5 + a sub 6 well that's just the same thing as the partial sum this is just the same thing as the partial sum of the first six terms for our infinite series it's just going to be the partial sum S sub 6 and we know how to algebraically evaluate what s sub 6 is we can apply this formula that we were given S sub 6 is equal to well everywhere we see an N we replaced with the 6 it's going to be 6 squared minus 3 over 6 to the third plus 4 and so what is this going to be 6 squared is 36 minus 3 so that's 33 and 6 to the 3rd let's see 36 times six I always forget my brain wants to say to 16 but let me make sure that that's actually the case 6 times 30 is 180 plus 36 yes it is 216 so I guess I have inadvertently by seeing 6 to the third so many times in my life I have inadvertently memorized six to the third power it's never never a horrible thing to have that in your brain so this is going to be to 16 plus 4 so 200 220 so S sub six or the sum of the first six terms of this series right over here is 33 220th and we're done and hopefully and the whole point of this is just so that you kind of appreciate or really do appreciate this partial sum notation and understand what it really means