Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    1
    ReputationRep:


    Doing an online test, can't leave until I finish this.
    I think this should be an application of reversing a Taylor series. There is no coefficient with the x, so this series should be about point x=0. And I figured out the sequences given below using the Taylor Series formula.


    f(0) = 1

    f'(0) = 1/8

    f''(0) = 9/64

    f'''(0) = 153/512

    f''''(0) = 3825/4096
    What I need to do next is find the equation f(x) which satisfy these sequences and I don't know how to continue it. Or this is not a Taylor series at all?
    Online

    22
    ReputationRep:
    (Original post by Y.X.)


    Doing an online test, can't leave until I finish this.
    I think this should be an application of reversing a Taylor series. There is no coefficient with the x, so this series should be about point x=0. And I figured out the sequences given below using the Taylor Series formula.


    f(0) = 1

    f'(0) = 1/8

    f''(0) = 9/64

    f'''(0) = 153/512

    f''''(0) = 3825/4096
    What I need to do next is find the equation f(x) which satisfy these sequences and I don't know how to continue it. Or this is not a Taylor series at all?
    It looks like the numerators are 8-fold factorials and the denominator is powers of 8.

    i.e: \displaystyle a_n = \frac{n!^{(8)}}{8^n} - not sure if that helps.
    Online

    22
    ReputationRep:
    Probably just simpler to look at is as:

    \displaystyle a_n = \frac{(1+8)(1 + 8 + 8) \cdots  (1 + 8n)}{(8+8)(8+8 + 8) \cdots ( 8 + 8n)}
    • Thread Starter
    Offline

    1
    ReputationRep:
    I figured out the denominator to be (8^n)x(n!). As for the 8-fold factorial, I don't even know how to express that in the test lol. I think the question is asking me to write a formula to represent the sum of the infinite terms of the series.
    (Original post by Zacken)
    It looks like the numerators are 8-fold factorials and the denominator is powers of 8.

    i.e: \displaystyle a_n = \frac{n!^{(8)}}{8^n} - not sure if that helps.
    Offline

    19
    ReputationRep:
    (Original post by Y.X.)
    I figured out the denominator to be (8^n)x(n!). As for the 8-fold factorial, I don't even know how to express that in the test lol. I think the question is asking me to write a formula to represent the sum of the infinite terms of the series.
    Your series will converge to (1 - x) to the power of minus 1/8, valid for -1 <x<1
    Online

    22
    ReputationRep:
    (Original post by TeeEm)
    Your series will converge to (1 - x) to the power of minus 1/8, valid for -1 <x<1
    Urgh, how did I not spot that. :facepalm:
    • Thread Starter
    Offline

    1
    ReputationRep:
    I checked this myself and it matched all the terms, thank you so much. Just can't find this type of question anywhere, maybe I need to search for conversion instead of Taylor series.
    (Original post by TeeEm)
    Your series will converge to (1 - x) to the power of minus 1/8, valid for -1 <x<1
    Offline

    19
    ReputationRep:
    (Original post by Y.X.)
    I checked this myself and it matched all the terms, thank you so much. Just can't find this type of question anywhere.
    you would in my undergrad resources under series, but it looks that someone very keen took them down while I was away.
    all the best
    Offline

    16
    ReputationRep:
    (Original post by TeeEm)
    Your series will converge to (1 - x) to the power of minus 1/8, valid for -1 <x<1
    :eek:
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    What's your favourite Christmas sweets?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.