Well, for starters, I could just be plain ol' wrong. But here's how I looked at different parts of the construction to try to make intuitive and visual sense out of the exapnsion, and what happen when you approach infinity... I'm going to walk through my logic step-by-step, so you can see how I got there (or top me right where I'm making a goof.)
First, to make sense of the 1/n part in the front, which can easily confuddle - your expansion will always have
n terms to sum... and each of them will be multiplied by 1/
n. So, for example, if
n is ten, you have ten terms, each multiplied by 1/10th. That means this:
View attachment 11149
It's like slicing the series into ten pieces and then adding them up. I can make better sense of what I'm "slicing up" if I move the 1/10th... like so:
View attachment 11150
As an example, if all of the ten terms on the top were a "1" - for example, if it were like this:
View attachment 11151
You should be able to easily see that it totals up to 1... Slicing
n ones into pieces of 1/
nth is like adding up
n ones and dividing by
n... which adds up to one.
Now, going back to the original problem, I'll write out the expansion of the summation in terms of
n... so for the first term, we replace
k with 0, and for the last, we replace
k with (
n-1).
View attachment 11157
What happens as n approaches infinity? Ignore the bottom n for now; maybe you see why I pulled it down there now, because if you leave it in front of each term it can confuse.
The first term:
View attachment 11158
Will always be 1 inside the radical. And the square root of one is one. So the first term will always be one.
The last term:
View attachment 11159
Will have the numerator get closer and closer to double the denominator. As
n goes to infinity, it will approach 2
n/
n... which is 2. So the last term becomes the square root of 2.
So the general, again:
View attachment 11160
when
n goes to infinity, will reduce to:
View attachment 11161
So you're covering the range from radical 1 to radical 2... but you're dividing it into thin slices. It's kinda like finding the average value between 1 and radical 2 - whihc woudl be (1 + 1.4142135 ) / 2, or 1.207107... which makes intuitive sense. It's a little bit more than one... when all the terms on top were a 1, add up ten million of them and divide by ten million, you get a 1. Instead, you're adding up ten million terms which go from 1 to 1.4142135, and then dividing by ten million. Makes sense? But it's not quite right...
Because that would only work if the numbers between radical 1 and radical 2 progressed linearly. They don't, because the radical function makes a curve, not a line. Going back to my terrible drawing:
That's the curve of the radical funcion f(
x) = SQRT(
x). (The line should get up to 2 when
x is 4, and should get up to 3 when
x is 9.)
But for us, only a small region matters.... at the beginning of the expansion, when
k is zero, the term simplifies to 1. And at the end, where
k gets as big as
n, it approaches 2. So that's the range I want to shade in, from 1 to 2. What's the top bound? The radical x function... so veriticall, I go up to radical (1) at the start, and up to radical (2) at the end.
Satisfyingly, this area is going to be a hair larger than my initial estimate of 1.207... I drew a horizontal line at
y = 1 for convenience; the shaded area below that has an area of 1. If the shade area extended up to the dotted line at 1.414, the area would be 1.414. If the curve from (1,1) to (2,1.414) were a straight line, then 1.207 would be correct - but it's not a straight line, it's a curve the bellies up. So the real answer is just over 1.207.
At this point, I could easily do the integral of the radical
x function from 1 to 2 to get the exact numerical answer... but I'm certain that this is NOT the approach the calc 2 prof would want to see. Even if I'm right, I'm getting at the answer the wrong way.