Don't worry about all of them at once; focus on just one term, \(\displaystyle a_n\). That is a finite sum. You just have to find a number that that one sum is less than. (It will turn out to be a number that they are all less than!) Give it a try. Follow my suggestion.
Do you know about the harmonic numbers \(\displaystyle H_n~?\) and the the digamma function. Here are two webpages One & Two.I want to show the above ^ Any ideas?
I understand that logically - but I don’t understand how I can actually go about proving this via implications. Everything you said makes sense but I am wanting to prove this thoroughly and my proof just looks messy and not thorough enough.No, just answer my questions! What is the largest term in the sum?
It may help (or may not) if you first consider a particular value of n, just to get a more human image of what this is. \(\displaystyle a_4\), for example, is 1/5 + 1/6 + 1/7 + 1/8. What is the largest term? If you replaced each term by that one, would the resulting sum be larger? Do the same thing to the general term, \(\displaystyle a_n\).
One way or another, I need to see you do some work!
I can see it is 1. But I don’t know how one can actually prove that it is 1...Your work is perfect but a little incomplete. n/(n+1) < what number for any positive number n
If you really do not see this, then plug in positive numbers for n and see what number than are all less than.
No, I disagree with you saying that you can see that n/(n+1) < 1. It sounds to me that you think that the only number that n/(n+1) is less than is 1. n/(n+1) is also less then 3/2, sqrt(7), pi and the number e.I can see it is 1. But I don’t know how one can actually prove that it is 1...
You seem to have confused me now. Is there any way you can explain this in a different angle?Surely you can show that n/(n+1) < 1 for all n. But you don't really need to do that.
Rather than use the fact that each term is no more than 1/(n+1), just use the fact that each term is less than 1/n!
There are two angles I've suggested: You can just continue what you did to get to the final answer, for which we've given you plenty of guidance; or you can back up and redo it, replacing 1/(n+1) with 1/n in your n terms. [I see Jomo just wrote about that, but got it wrong - don't let that confuse you.] You don't need to attempt the latter if you don't see my point. Just finish the work you did, and show us your conclusion.You seem to have confused me now. Is there any way you can explain this in a different angle?