Suppose ∑ n = 1 ∞ x n < ∞. Then prove that ∑ n...

Harmony Oneal

Harmony Oneal

Answered

2022-11-25

Suppose n = 1 x n < . Then prove that n = 1 x n y n converges.

Answer & Explanation

Cale Terry

Cale Terry

Expert

2022-11-26Added 10 answers

Let N 2 then by telescoping
y N = y 1 n = 1 N 1 ( y n y n + 1 )
hence the sequence ( y N ) is convergent since
lim N n = 1 N 1 ( y n y n + 1 ) = n = 1 ( y n y n + 1 ) <
by M>0 and by
inf ( M x n , M x n ) x n y n max ( M x n , M x n )
and n = 1 x n is convergent we have the result.

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