Integral Test for Convergence Using Finite Sum

Question

I need some help understanding what this question is asking me to do.

Do we calculate a finite sum for some arbitrary l and then calculate the integral? How do we apply this estimate to the limit?

Answer 1

When you are evaluating a series $\sum\limits_{i=1}^{\infty}a_{i}$ You can think of it as summing the areas of rectangles with width $1$ and height $a_i$. Effectively you have a Rectangular Approximation of an Integral, like so (not exact, but works):

So as you can see this Rectangular Approximation is an underestimate of the integral of the function. Meaning, if the integral of the function to infinity converges, then the series representing the sum of the Rectangular Approximations must also converge. Given, that the Approximation is an underestimate. The Rectangular Approximation for $f(x)$ from $1$ to $\infty$ with rectangles of width 1 would be $\sum\limits_{i=1}^{\infty}f(i)$. Now if $f(x)$ is integrable, and is approaching $0$, and the sequence $f(i)$ is either always positive or negative. If you evaluate $\int\limits_{1}^{\infty}f(x)dx$ to be a number, then the series must converge because as discussed earlier, the series is an underestimate (or overestimate if the series is always negative) of the actual integral so then, if all the terms of the series is always positive (or always negative) then the series would evaluate to a value between the integral and $0$, and so the series converges.

If the integral diverges however, we can think of the the Riemann sums,and hence series, to be an overestimate (underestimate in the negative case) by reindexing the series. So if the integral diverges (underestimate in the negative case), and the series is an overestimate, then the series diverges.

So the problem is asking you to evaluate, for each of the series $\sum\limits_{i=2}^{\infty}f(i)$, the integral $\int\limits_{2}^{\infty}f(x)dx$ and use the convergence of the integral to determine the convergence of the series.