Hello maths
Suppose m and n are positive integers such that
⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ S = k = 0 ∑ n ( − 1 ) k k + m + 1 1 ( k n ) T = k = 0 ∑ m ( − 1 ) k k + n + 1 1 ( k m ) .
Find the value of S − T .
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1 solution
@Alan Yan , Sorry i am unable to understand how you converted summation to integral. Can you please elaborate your step 1 more clearly?
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Try expanding the polynomial inside the integral and integrating each individual term. You'll find that it will be just the difference of the two sums.
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That is true. I am asking how you converted summation sign into integral one?
It is clear that S − T = ∫ 0 − 1 ( − 1 ) m + 1 ( 1 + x ) n x m − ( − 1 ) n + 1 ( 1 + x ) m x n d x . But this is equal to 0 since ∫ 0 − 1 ( 1 + x ) m x n d x = ( − 1 ) m + n ∫ 0 − 1 ( 1 + u ) n u m d u after the change of variable u = − 1 − x .