November 14th, 2017, 08:31 AM  #1 
Member Joined: Jan 2016 From: Blackpool Posts: 95 Thanks: 2  Convergence question
for part one of the question it asks me to find the radius of convergence of the series \[\sum_{n=0}^{\infty}\frac{x^n}{n}\] and i got the answer x<1 so radius of convergence R=1 for the second part of the question I am asked to show that whenever x<R , f'(x)=1/(1x) for all such x. Could anyone give me a hint to the second part and also check if i have done the first part correct, thanks! 
November 14th, 2017, 09:04 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,011 Thanks: 1044 
the idea here is that provided the series converges it represents the integral of $f^\prime(x) = \dfrac{1}{1x}$ This is easily integrated to $f(x) = \ln(1x)$ So you have to show that $\sum \limits_{n=1}^\infty \dfrac{x^n}{n} = \ln(1x)$ can you do that? note: you have the series incorrectly starting at $n=0$ The 0th term is $ln(10) = 0$ 
November 14th, 2017, 10:24 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,331 Thanks: 2457 Math Focus: Mainly analysis and algebra 
I would think that you could demonstrate that $f'(x)= \frac1{1x}$ by differentiating term by term and then demonstrating that the result is equivalent to the MacLaurin series for $\frac1{1x}$.

November 15th, 2017, 04:13 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,248 Thanks: 887 
Or simply observing that $\frac{1}{1 x}$ is the sum of the geometric series $\displaystyle 1+ x+ x^2+ x^3+ \cdot\cdot\cdot$. as long as x< 1.

November 15th, 2017, 02:44 PM  #5  
Member Joined: Jan 2016 From: Blackpool Posts: 95 Thanks: 2  Quote:
Hi I used the maclaurin series to prove that the sum on x^n/n is approximately equal to ln(1x) and it works out when n>or equal to 1, thanks for the help guys.  

Tags 
convergence, question 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Weak Convergence Question  PeterPan  Real Analysis  1  January 21st, 2014 02:16 PM 
uni level convergence/limit question  RedPanda624  Real Analysis  2  December 3rd, 2013 12:34 PM 
Series Question  Convergence  Anthony28  Real Analysis  1  April 12th, 2012 12:00 PM 
Another series question on uniform convergence.  watson  Real Analysis  1  December 31st, 2011 01:02 PM 
Question regarding uniform convergence  Anon123  Real Analysis  1  September 1st, 2010 01:37 PM 