April 11th, 2018, 05:05 AM  #1 
Member Joined: Apr 2018 From: On Earth Posts: 34 Thanks: 0  Infinite Sums
T = 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 The numerator of each subsequent term is the sum of the numerators of the previous two terms. The denominator of each subsequent term is twice the denominator of the previous term. 1) By considering the first six terms of 1/2 T and 1/4 T, find the values of a, b, c, d, e and f where 3/4 T = a/4+ b/8 + c/16 + d/32 + e/64 + f/128. 2) Hence find the value of T. For 1) I tried doing 1/2 T and 1/4 T and adding them together, but I am not sure whether that is the correct method. Please help! Thanks. Last edited by skipjack; April 11th, 2018 at 05:46 AM. 
April 11th, 2018, 05:40 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
Looks like the thing to try. What did you get?

April 11th, 2018, 05:44 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,926 Thanks: 2205 
T $\ \ \ \ $ = 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + . . . T/2 $\ $ = 1/4 + 1/8 + 2/16 + 3/32 + 5/64 + 8/128 + . . . T/4 $\ $ = $\ \ \ \ \ $ 1/8 + 1/16 + 2/32 + 3/64 + 5/128 + 8/256 + . . . 3T/4 = 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + 13/128 + . . . = T  1/2 Hence T = 2. 
April 11th, 2018, 06:39 AM  #4  
Senior Member Joined: Oct 2009 Posts: 850 Thanks: 326  Quote:
 
April 11th, 2018, 06:48 AM  #5 
Senior Member Joined: Oct 2009 Posts: 850 Thanks: 326 
A method that also shows convergence is to represent the numerators (the Fibonnaci numbers) by their formula: $$\sum \frac{F_0}{2^n} = \sum \frac{\varphi^n  (\varphi)^{n}}{2^n \sqrt{5}} = \frac{1}{\sqrt{5}}\sum \left(\frac{\varphi}{2}\right)^n  \frac{1}{\sqrt{5}} \sum\left(\frac{1}{2\varphi}\right)^n$$ Using geometric series, we get $$\frac{1}{\sqrt{5}}\frac{1}{1 (\varphi/2)}  \frac{1}{\sqrt{5}}\frac{1}{1+(1/(2\varphi)} $$ Some tedious algebra shows that this indeed evaluates to 2. 

Tags 
infinite, sums 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Infinite intersections and infinite unions  Azzajazz  Real Analysis  5  March 10th, 2016 08:01 PM 
Another question about sums  Carl James Mesaros  PreCalculus  6  February 27th, 2015 06:12 AM 
Relation between an infinite product and an infinite sum.  Agno  Number Theory  0  March 8th, 2014 04:25 AM 
Infinite set contains an infinite number of subsets  durky  Abstract Algebra  1  March 15th, 2012 11:28 AM 
Infinite series  finding sums  xdeathcorex  Calculus  4  August 31st, 2010 11:03 AM 