Question

find the sum of the sigma ? question attatched PLEASE HELPPPP! WILL MEDAL

Answer 1

First\[\sum_{n =1}^{\infty}\frac{ 10n ^{2} }{ 3 }=\frac{ 10 }{ 3 }\sum_{n =1}^{\infty}n ^{2}\]Now\[\sum_{i =1}^{k}=\frac{ k(k +1)(2k +1) }{ 6 }\]Use this identity to help you further simplify.

Answer 2

Once you have simplified find the limit as k "approaches" infinity, to find the desired sum.

Answer 3

\(\Large \sum_{n =1}^{k}n^2=\frac{ k(k +1)(2k +1) }{ 6 }\)

Answer 4

Yes @mathmate thank you for bringing my attention to the typo! I meant to write n^2 after the summation.

Answer 5

@mathmate So would your answer be the answer?

Answer 6

No, I just gave you a hint and a formula to get you started.

Answer 7

well im not quite sure how to do it.

Answer 8

@sar12389
@calculusfunctions has given you sufficient hints to get you started.
If you reread what he wrote, you will find your answer!

Answer 9

so would it be infinity over 6 ?

Answer 10

Okay @sar12389 I'll start you off with the first step.\[\sum_{n=1}^{\infty}\frac{ 10n ^{2} }{ 3 }\]\[=\frac{ 10 }{ 3 }\sum_{n =1}^{\infty}n ^{2}\]\[=\frac{ 10 }{ 3 }\lim_{k \rightarrow \infty}\frac{ k(k +1)(2k +1) }{ 6 }\]Now evaluate the limit and multiply the result by 10/3.

Answer 11

how do you evaluate the limit? @calculusfunctions

Answer 12

Since the degree of the numerator is greater than the degree of the denominator, the limit is infinite.

Answer 13

Do you see what the answer is now?

Answer 14

Since the limit is infinite, the sum will be infinite. Therefore the given series is divergent and does not have a finite sum.

Answer 15

ohh okay

Answer 16

So do you understand @sar12389

Answer 17

i think so, so would this also have an infinite sum ? @calculusfunctions

Answer 18

not have **

Answer 19

\[\sum_{n =1}^{\infty}(r)^{n}=\frac{ r }{ 1-r }\]If -1 < r < 1

Answer 20

\[\sum_{n=1}^{infinity}(\frac{ 1 }{ 4 })^n\]

Answer 21

but where would r come in

Answer 22

What do you mean? r is the common ratio. In your question, r = 1/4.

Answer 23

oh okay. so would that be the answer? what you said?

Answer 24

What would be the answer?? Now that you know r = 1/4, you have to substitute for r in the formula (right side of the equation) I gave you to find the final answer.

Answer 25

(1/4)^n=1/4/1-1/4 =.25/.75 =0.3333... ?

Answer 26

Yes, that is correct! 1/3 is the answer.

Answer 27

and for the first question, the answer would be, "it does not have a finite sum"? @calculusfunctions

Answer 28

Yes, the first series was divergent (sum does not exist as a finite number). The second series is convergent (sum exists as a finite number).

Answer 29

THANK YOU SO MUCH!!!!