Suppose ∑ n = 1 ∞ x n &lt; ∞. Then prove that ∑ n = 1 ∞ x n y n converges.

Question

Suppose       ∑          n      =      1              ∞                  x      n        &amp;lt;  ∞. Then prove that       ∑          n      =      1              ∞                  x      n              y      n       converges.

Cale Terry · Accepted Answer

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              )    &amp;lt;    ∞  by M&amp;gt;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.

Suppose sum_(n=1)^infty x_n< infty, sum_(n=1)^infty |y_n-y_(n+1)|<infty. Then prove that sum_(n=1)^infty x_ny_n converges.

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