# Sequences and sums

• Apr 12th 2010, 01:27 PM
charikaar
Sequences and sums
Suppose that $\displaystyle (x_k)_{k=1}^{\infty}$ and $\displaystyle (y_k)_{k=1}^{\infty}$ are sequences of positive real numbers, and that $\displaystyle \sum_{k=1}^{\infty}x_{k}=S$ and $\displaystyle \sum_{k=1}^{\infty}y_{k}=T$. Let $\displaystyle z_k=\max(x_k,y_k)$. Prove that the sum $\displaystyle \sum_{k=1}^{\infty}z_k$ exists.

Let $\displaystyle S_n=\sum_{k=1}^{\infty}x_{k}$ and let
Let $\displaystyle T_n=\sum_{k=1}^{\infty}y_{k}$

We know $\displaystyle S_n$ coverges to S and $\displaystyle T_n$ converges to T. So $\displaystyle S_n+T_n$ converges to S+T. Am i on the right track? I can't go any further from here so any help would be greatly appreciated.

Thanks
• Apr 12th 2010, 01:56 PM
awkward
Quote:

Originally Posted by charikaar
Suppose that $\displaystyle (x_k)_{k=1}^{\infty}$ and $\displaystyle (y_k)_{k=1}^{\infty}$ are sequences of positive real numbers, and that $\displaystyle \sum_{k=1}^{\infty}x_{k}=S$ and $\displaystyle \sum_{k=1}^{\infty}y_{k}=T$. Let $\displaystyle z_k=\max(x_k,y_k)$. Prove that the sum $\displaystyle \sum_{k=1}^{\infty}z_k$ exists.

Let $\displaystyle S_n=\sum_{k=1}^{\infty}x_{k}$ and let
Let $\displaystyle T_n=\sum_{k=1}^{\infty}y_{k}$

We know $\displaystyle S_n$ coverges to S and $\displaystyle T_n$ converges to T. So $\displaystyle S_n+T_n$ converges to S+T. Am i on the right track? I can't go any further from here so any help would be greatly appreciated.

Thanks

You might be able to make use of the fact that $\displaystyle z_k < x_k + y_k$.