We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
168
1
avatar

 

Let a and b be positive real numbers, with a>b. Compute \(\frac{1}{ba} + \frac{1}{a(2a - b)} + \frac{1}{(2a - b)(3a - 2b)} + \frac{1}{(3a - 2b)(4a - 3b)} + \dotsb.\)

 Dec 8, 2018
 #1
avatar+5226 
+1

Take a subsequence of N terms of this.

 

Split each term up using partial fractions

 

You'll see this reveals that the sequence is a telescoping sequence.

 

After the sum only part of the Nth term remains

 

\(\dfrac{N}{b (a N+b (-N)+b)}\)

 

Taking the limit of this as N goes to infinity results in

 

\(\dfrac{1}{b(a-b)}\)

.
 Dec 8, 2018

17 Online Users

avatar