Want to understand how a fraction is simplified The fraction is used to determine the sum of a telescopic series. sum_(k=1)^oo 1/((k+1)sqrt(k)+k sqrt(k+1))

sdentatoiz 2022-09-13 Answered
Want to understand how a fraction is simplified
The fraction is used to determine the sum of a telescopic series.
k = 1 1 ( k + 1 ) k + k k + 1
This is the solved fraction.
1 ( k + 1 ) k + k k + 1 = ( k + 1 ) k k k + 1 ( k + 1 ) 2 k k 2 ( k + 1 ) = ( k + 1 ) k k k + 1 k 3 + 2 k 2 + k k 3 k 2 = ( k + 1 ) k k k + 1 k ( k + 1 ) = k k k + 1 k + 1 = 1 k 1 k + 1
I want to understand what is being done on every step,mainly the last three. Thank you.
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Answers (2)

Ashlee Ramos
Answered 2022-09-14 Author has 20 answers
First equal: multiply and divide by ( k + 1 ) k k k + 1
Second equal: Expand the denominator
Third equal: cancel obvious terms in the denominator and write k 2 + k = k ( k + 1 ).
Fourth equal: distribute along the minus in the numerator, and cancel the obvious factors ( k + 1 ) in the first summand, and k in the second one:
( k + 1 ) k k k + 1 k ( k + 1 ) = ( k + 1 ) k k ( k + 1 ) k k + 1 k ( k + 1 ) = k k k + 1 k + 1 .
Fifth equal: k k = 1 k , and similarly for k + 1

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tuzkutimonq4
Answered 2022-09-15 Author has 1 answers
Another way to look at it is first "declutter" the expression, since all those radicals obfuscate the simple structure. Let a = k , b = k + 1 , then:
1 ( k + 1 ) k + k k + 1 = 1 a b 2 + a 2 b = 1 a b ( a + b )
Now consider that ( b + a ) ( b a ) = b 2 a 2 = ( k + 1 ) k = 1 , so 1 a + b = b a . Then:
1 a b ( a + b ) = b a a b = b a b a a b = 1 a 1 b = 1 k 1 k + 1

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