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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
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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

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