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(p/(1-(x^p)))-(q/(1-x^q))) where li tends to 1 and p and q are natural number

(p/(1-(x^p)))-(q/(1-x^q))) where li tends to 1 and p and q are natural number

Grade:12

2 Answers

Anoopam Mishra
126 Points
5 years ago
p/(1-x^p) - q/(1-x^q)
(p-px^q-q+qx^p)/(1-x^p-x^q+x^{p+q})
At x tending to 1, it is of indeterminant form 0/0.
Using L-Hospital rule
 
Anoopam Mishra
126 Points
5 years ago
(-pqx^{q-1}+pqx^{p-1})/(-px^{p-1}-qx^{q-1}+(p+q)x^{p+q-1})
This is again of 0/0 form.
Again use L-Hospital rule
(-pq(q-1)x^{q-2}+pq(p-1)x^{p-1})/(-p(p-1)x^{p-2}-q(q-1)x^{q-2}+(p+q)(p+q-1)x^{p+q-2})
Now putting x=1.
(pq(p-q))/(-p^2-q^2+p+q+p^2+q^2+2pq-p-q]) = (p-q)/2
 

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