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Prove part (i) of Theorem 8.

$\sum_{n=1}^{\infty}\left(c a_{i}\right)=c\left(\sum_{n=1}^{\infty} a_{n}\right)$

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Campbell University

Harvey Mudd College

Idaho State University

Boston College

Let's prove par one of theory me in this section and this says If the sum of am converges, then thes to terms are equal here, we're assuming, of course, see is just a constant. So fix real number. So So show these air equal. Well, let's look at the left hand side. So let's call U N the end partial some of the left hand side. So the left hand side. We know that we can always rewrite the whole sum as the limit of partial sums. So you and usually you see a written in this form, the whole son from one to infinity equals the limit of s end. So here, in our case, we have C and this is the entire sum going all the way up to infinity. Starting point might be one. Maybe not. But since it's not given much, just call it something. Let's call it one we can rewrite. This is a limit of the partial sums here, but still, actually, let me not call it s because we're using you. This's UK, The partial sums. So let's rewrite that partial some. So here we can rewrite UK as C A one si a Tu all the way up to see a k k partial some. This's just by definition of uk of here partials on then. Since I'm just dealing with a finite song, I could go ahead and factor out that see, And then I could also pull the si in front of the limit. This's just by one of your limit properties. So this is just using, ah, limit their, um, when you first learned about limits and then here let's go to the next page a one all the way up to a K. However, this now is just sn where's on the right hand side? We had this term here and then we're calling us in the sum of a one, as usual, up to an and then Therefore we could go ahead and replace The limit of SN was just the entire sum of a N, and that's exactly what we wanted to prove. So this is our final answer. The whole argument