20040809, 01:53  #1 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Applying the Binomial Theorem More Than Once
I already know how to expand an expression like sqrt(b+c) into an infinite series using the binomial theorem.
But what if I have to apply the process again? This time, I am trying to expand an expression that already has an infinite number of terms. The simplest example of this would be sqrt(a+sqrt(b+c)) Thanks 
20040810, 17:28  #2  
Nov 2003
16444_{8} Posts 
Quote:
sqrt(a + d). What else might you want? Please specify. 

20040811, 13:26  #3  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Applying the Binomial Theorem more than once
Quote:
As I understand it that if we put b+c = d then what is meant is that the value of the expression sqrt(a+sqrt.d) is required. This is a surd (irrational no.) and does not need the Binomial Theorem for its solution. If a straight forward value of sqrt (a + Sqrt (b+c) ) is required assuming that a,b,c,d, are natural nos. then the theory, method, and solution can be provided by me Mally 

20040812, 01:07  #4 
Dec 2003
Hopefully Near M48
6DE_{16} Posts 
If I use Bob Silverman's suggestion, I will end up with an infinite series where each term is itself an infinite series. Is that supposed to happen?

20040812, 10:25  #5  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
10101100011001_{2} Posts 
Quote:
Now multiply out the series, dropping those terms which have an exponent larger than those in which you are interested. Paul 

20040812, 16:58  #6  
Bronze Medalist
Jan 2004
Mumbai,India
2052_{10} Posts 
Applying the binomial theorem more than once
Quote:
Excepting the first termYes After the 18t century mathematicians were forced to break away from the ancient Greek practice of picturing formulae as in their geometry, which explains the comparative stagnation for 2000 years till modern maths arrived on the scene. Please don’t fall into the same error. I give below a worked example and the method used. The propositions I mention can be proved. If required please consult a good text book on elementary Algebra on surds (irrationals) Eg: Find sqrt.(10 + 2 sqrt. 21)= (A) say, Let (A) be = sqrt. x + Sqrt. yProposition (1) Then ( sqrt 10  2sqrt. 21 = sqrt. x – sqrt y ) Proposition(2) By multiplication Sqrt ( 100 – 84 ) = x  y Therefore 4 = x – y (B) By squaring (A) we get 10 + 2 sqrt. 21 = x + y + 2 sqrt ( x* y ) By equating rational partsProposition ( 3 ) We get x + y = 10 From (B) x  y = 4 Therefore x = 7 ; y = 3 Hence (A) =sqrt 7 + sqrt 3 Any difficulty please let me know. Mally 

20040819, 17:29  #7 
Bronze Medalist
Jan 2004
Mumbai,India
2052_{10} Posts 
Applying the Binomial Theorem more than once
Quote:Originally Posted by jinydu If I use Bob Silverman's suggestion, I will end up with an infinite series where each term is itself an infinite series. Is that supposed to happen? unquote If you still insist on the Binomial Theorem derivation try solving this problem Simplify: sqrt (1+ sqrt[1a^2]) + Sqrt (1 sqrt [1a^2]) Hint: both terms are related thus: sqrt(x) +sq rt (y) and sqrt(x)Sqrt(y) Ans: sqrt (2[1+a]) Try it by the method I have given Mally 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Fast calculation of binomial coefficients  mickfrancis  Math  4  20160815 13:08 
Binomial Primes  Lee Yiyuan  Miscellaneous Math  31  20120506 17:44 
No Notice Binomial Coefficients, Pascal's triangle  Vijay  Miscellaneous Math  5  20050409 20:36 
I need a proof for this binomial property.  T.Rex  Math  3  20041008 19:13 
Binomial Expansion Applet  jinydu  Lounge  2  20040505 08:33 