Question

Can someone please help me find the sum of this sigma notation?

\[\sum_{k=1}^{10}2k^3\]

Answer 1

Alright, this one is not too bad... Do you know how to interpret the sigma notation?

Answer 2

no :(

Answer 3

OK, here's a quick review of how to read sigma notation. Let's use an easy first question. \[\sum_{k=1}^{3} k\]

Answer 4

So what the weird sigma thing means is to add up a bunch of numbers.

Answer 5

the bottom part (k=1) tells you what to plug in first, and the top part (3) tells you when to stop.

Answer 6

So, we just get \[\sum_{k=1}^3 k = (1) + (2) + (3) = 6\]

Answer 7

so keep plugging in numbers until you get 3?

Answer 8

OH i see!!

Answer 9

does that make sense?

Answer 10

great! so for the question you were trying to answer....

Answer 11

the first thing is going to be 2(1)^3

Answer 12

then, we need to add 2(2)^3...

Answer 13

then, 2(3)^3 and so on

Answer 14

it's a bit long, but not too hard... :)

Answer 15

I gotcha! thank you SO much for taking the time to explain everything to me, I really appreciate it! :)

Answer 16

no problem!

Answer 17

so to find the "sum" when I get 2(1)^3...2(10)^3 do i add all of them up?

Answer 18

that's right, you should get a pretty big number... if I did the adding right, you should get 6050

Answer 19

thank you! :)

Answer 20

yw

Answer 21

@jtvatsim for this problem instead of starting at 1 would i start at 0??

\[\sum_{j=0}^{4}(j^2+1)\]

so it would be (0)^2+1, (1)^2 +1, (2)^2+1, (3)^2+1 and (4)^2 +1?

Answer 22

you've got it! You're on a roll now. :)

Answer 23

Sigma notation just looks big and scary, but it's really actually nice. lol

Answer 24

thanks so much! no kidding :p

Answer 25

so for this one

\[\sum_{k=1}^{\infty}8(\frac{1}{10})^k\] how would i solve this??

Answer 26

OK, this one is a bit different since using the method we've been doing would require us to find an infinite amount of numbers (no thank you...)

Answer 27

Instead, let's try to get an idea of what is going on by writing out just a few terms...
\[\sum_{k=1}^\infty 8(1/10)^k = 8(1/10) + 8(1/10)^2 + 8(1/10)^3 + ...\]

Answer 28

of course, we have to keep going forever, but notice that we can factor out the 8 showing us that we really have \[8[ (1/10) + (1/10)^2 + (1/10)^3 + ... ]\]

Answer 29

OK, great, but now what??? Well, notice that this is a geometric series since we are just multiplying by (1/10) each time.

Answer 30

The geometric series formula is \[\frac{ a }{ 1-r }\] where a is the first term and r is the ratio.

Answer 31

so even though we know it is multiplying by 1/10 each time is there a way to "figure out" infinity?

Answer 32

Yes, that's what the geometric series formula does... it adds up infinitely many things that multiply by some ratio each time.

Answer 33

So, it turns out (by the geometric series formula) that
\[[(1/10) + (1/10)^2 + (1/10)^3 + ...] = \frac{ 1/10 }{ 1-(1/10) } = 1/9\]

Answer 34

Now, remember that we still had the 8 out front so the final answer is 8*(1/9) = 8/9

Answer 35

so 8/9 would be infinity? in this equation at least?

Answer 36

Hmm, not quite. What the sigma was asking us was to start with k=1 and keep adding FOREVER! You might think that if you add forever you would never get a "final" answer. However, math is a bit surprising, so what we have found (using the tricky geometric series) is that if we were to add these numbers FOREVER (the infinity) we would actually get the answer 8/9. Does that make sense? Infinity tends to be hard to imagine. :)

Answer 37

oh okay, that makes more sense to me now :P you're awesome! i wish i could give you an infinite amount of medals :D thanks again :)

Answer 38

LOL, that would be nice, but that might take up all the memory on the internet! :D

Answer 39

hahaha!