Question

(Calculus) Need help with finding a function represented by the series!

Answer 1

I've attached the question.

Answer 2

I can see how it might translate into an ln(x) function, but I don't really know how to do it in a "concrete" manner.

Answer 3

show me what you can "see"

Answer 4

Well, I know that Ln(sqrt(x)) is equal to 1/2 Ln(x). So, that would account for the 1/2 factored outside of the summation of the series.

Answer 5

Note: \[\sum_{n=0}^{\infty} a x^n \implies \frac{ a }{ 1-x }, |x|<1\]

Answer 6

this is what i see when i try to post to any question now ...

Answer 7

Ah I guess I'll let nin help with this one :)

Answer 8

Yes, I know the 1/(1-x) formula as well. I don't see how to manipulate it to work for this problem however.

Answer 9

and if this is not a trial run of the QH stuff, i apologoze for the irrelevant post :)

Answer 10

I don't see the attachment.

Answer 11

hmm, let me repost it

Answer 12

Let me know if you got it.

Answer 13

yes, I did get it

Answer 14

alright, show me what your solution is and yes, the ln comes up

Answer 15

Hmm, how did you get to that specific answer though (if you don't mind explaining)?

Answer 16

well, you saw what batman put in terms of general formula and to obtain k from algebra, we need to undo it by using ln

Answer 17

So, I would have Ln(1-x)? I'm having a hard time seeing what the "X" term would be in this problem.

Answer 18

yes

Answer 19

you have x^(10k)

Answer 20

-> x^10

Answer 21

is there any particular reason that the "k" is removed from the x^10 term?

Answer 22

(might be a dumb question, sorry haha)

Answer 23

There are no dumb questions!

Answer 24

(-:

Answer 25

:)

Answer 26

And since no one else wants to welcome you...

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Hey there!!! Since you are new here, read this legendary tutorial for new OpenStudiers!! http://goo.gl/5pp1u0

@sabirrobby Welcome to the OS community, and congrats on being one of the few people to have asked for Qualified Help! :D

Answer 27

I think it would help me see what's happening if you could to explain how \[1/k \times10^k \] turns into x^10

Answer 28

I just tried to check my answer again and I was wrong

Answer 29

wolfram says it's negative but I have positive

Answer 30

@iambatman help a brotha out

Answer 31

You're almost there ! Integrating both sides brings us the required "k in the bottom

Answer 32

\(\color{blue}{\text{Originally Posted by}}\) @nincompoop
@iambatman help a brotha out
\(\color{blue}{\text{End of Quote}}\)

You can always ask the other Qualified Helper -- @perl

Answer 33

@perl help a nincompoop out

Answer 34

Tough question? ^-^

Answer 35

From converging geometric series we have \[\large \begin{align}\sum\limits_{k=\color{blue}{0}}^{\infty}x^{10k} &= \dfrac{1}{1-x^{10}} \implies \sum\limits_{k=\color{blue}{0}}^{\infty}x^{10k+9} &= \dfrac{x^9}{1-x^{10}} \\~\\
\end{align}\]

Integrate both sides and get


\[\large \begin{align} \int \sum\limits_{k=\color{blue}{0}}^{\infty}x^{10k+9} dx &= \int \dfrac{x^9}{1-x^{10}}dx\\~\\
\sum\limits_{k=\color{blue}{0}}^{\infty}\dfrac{x^{10k+10}}{10k+10} &= -\frac{1}{10} \ln (1-x^{10}) + C\\~\\
\sum\limits_{k=\color{blue}{1}}^{\infty}\dfrac{x^{10k}}{k} &= -\ln (1-x^{10}) + C\\~\\

\end{align}\]

see if you can take it home from here

Answer 36

So, when I factor in the -1/2 and get (ln(1-x^10))/2; I can convert that into \[\ln(1-x^10)^2\], right?

Answer 37

should be x^10 *

Answer 38

Kinda.. but I would multiply \(-\dfrac{1}{2}\) both sides instead
Also you need to find the value of \(C\) and get the final unique function

Answer 39

Would you mind showing me how to do that (sorry I'm really shook up by this question)?

Answer 40

start by multiplying -1/2 both sides
im still thinking about how to find the constant C

Answer 41

Well what is the domain?

Answer 42

\[\begin{align}
\sum\limits_{k=\color{blue}{1}}^{\infty}\dfrac{x^{10k}}{k} &= -\ln (1-x^{10}) + C\\~\\
-\frac{1}{2}\sum\limits_{k=\color{blue}{1}}^{\infty}\dfrac{x^{10k}}{k} &= \ln \sqrt{1-x^{10}} + C_1\\~\\
\end{align}\]

Answer 43

domain ? could you elaborate/provide a less cryptic hint batty

Answer 44

Haha, well I was thinking as in what is the restricted domain, we can then use those values of x, and find C.

Answer 45

Okay, thanks a lot for the help!

Answer 46

I see that works perfectly ! we can simply plugin x = 0 and solve C

Answer 47

Right :)

Answer 48

Is it alright if I close the question now?

Answer 49

You can close it when ever you like! I hope everything helped, take care and thanks for the question!

Answer 50

Alright, thanks a BUNCH you guys!