Turn on thread page Beta
    • Thread Starter
    Offline

    14
    ReputationRep:
    How do I prove that

    f'(x) =\displaystyle\sum_{n=1}^{\infty  } \frac{n x^{3n-1}}{3^{n-1} n!}

    is equal to

    x^2 f(x) =\displaystyle\sum_{n=0}^{\infty  } \frac{x^{3n+2}}{3^n n!}
    Offline

    18
    ReputationRep:
    I'm not sure what you actually mean by what you've just written.

    The two are not logically equivalent: the 2nd equation implies a specific value for f(1), while the 1st equation only describes the derivative and so f is not unique (adding a constant to f will not stop it satisfying the 1st equation).
    • Thread Starter
    Offline

    14
    ReputationRep:


    This is what i mean, sorry
    • PS Helper
    Offline

    14
    PS Helper
    Replace n by m+1 in the first series and rearrange stuff. Things will cancel.
    Offline

    20
    ReputationRep:
    (Original post by Dagnabbit)


    This is what i mean, sorry

    If you wanted to you could it like this:

    Rearrange the equation a bit to get
    f(x) =\displaystyle\sum_{n=0}^{\infty  } \frac{(x^3/3)^n}{n!}

    Notice that by the series definition of an exponential
    f(x) = \exp(\frac{x^3}{3})

    This implies that
    \ln f(x) = \frac{x^3}{3}

    Differentiate both sides with respect to x (need the chain rule for the LHS)
    \frac{f'(x)}{f(x)} = x^2

    Multiply both sides by f(x)
    f'(x) = x^2 f(x)

    Which is what you wanted
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: March 27, 2011

University open days

  • Staffordshire University
    Everything except: Midwifery, Operating Department Practice, Paramedic Undergraduate
    Sun, 21 Oct '18
  • University of Exeter
    Undergraduate Open Days - Exeter Campus Undergraduate
    Wed, 24 Oct '18
  • University of Bradford
    Faculty of Health Studies Postgraduate
    Wed, 24 Oct '18
Poll
Who is most responsible for your success at university
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

Equations

Best calculators for A level Maths

Tips on which model to get

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

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

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