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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 11: Estimating infinite series

# Series estimation with integrals

Suppose we want to find the sum of a convergent series, and can't do it directly. We can take a partial sum, but how do we know how far we are from the actual sum? We can use improper integrals for that!

## Want to join the conversation?

• at ,he says that the underestimate consists of the rectangles that are contained within the graph.So,shouldn't Rk be GREATER than the under estimate?Because the area under the graph is greater than the area of the rectangles contained in the graph.
• Sal means that Rk is an underestimate of an integral of the function. In other words Rk is less than the area under some part of the curve. But Rk is always the sum of the rectangles. It's greater than the area under one part of the curve, but less than the area under a different part of the curve.
• Since on the over-estimate we're starting the area/integral from k+1, shouldn't the Sk be S(k+1)?
• At , I am confused. If he chose k+1 as the height, why does the rectangle fill in back to 'k'? Isn't it suppose to go to 'k+2', thus making it an over estimation?
• You're right, if he made the rectangle width start at k+1, and end at k+2 instead, he would have an overestimation of the integral from (k+2) to infinity. However at , the time you mentioned, he was show trying to show an underestimation, so he ended the rectangle width at k. At he does exactly what you mention and creates an overestimation instead.
(1 vote)
• Sal can say this is an upper bound because, based on the sketch he drew, the particular riemann trapezoidal estimation was an underestimate. But this might not always be true, right? The estimate could sometimes be an overestimate. So I don't understand what to do from there:p
(1 vote)
• That is exactly why he uses the rectangles (not trapezoids) relative to both k and k+1: in one direction the smooth curve is an overestimate relative to the rectangles (which are the true value), and in the other direction the smooth curve of the integral is an underestimate relative to the rectangles (which are the true value). Once he has the underestimate AND the overestimate, he knows he has the true value of the infinite sum trapped in the middle like a firefly between your hands. :)
• at , why does he have to move over one interval to the right to make the rectangles? Also, does the rectangle from the interval from 0 to k represent S_k?
• Can someone explain why at Sal shows the series as the sum of the partial sum of the function, with the first one being n=1 to K, but the second one is n=k+1 to infinity? I'm more confused on the second part of that, but also how he re-wrote the series as a sum of a series and a partial sum
• At approx , Sal begins talking about the upper bound. Instead of using k+1 as the beginning of the bounding series, why not just take the left hand sum of the series? I'm having a hard time seeing how the series is an over estimate if there is a giant hole in this sum from k to k+1. Is there some difficulty in just taking the left hand sum?
• It seems our "overestimation" based on R_k+1 @ is smaller than our "underestimation" based on R_k @? How can this be?
Well, @ we are over estimating the smaller integral starting at k+1. And @ we are underestimating the larger integral starting at k. But the way it is explained here it seems we are taking over and under estimates of the same integral by shifting the Riemann approximation.
(1 vote)
• Is this topic (Series Convergence and Estimation) supposed to come 3rd in the Integral Calculus mission after Sequences and Series Intro & Mission Foundations?
I feel like it's assumed I know a lot of the subject matter that's a basis for being discussed in this topic but it's more or less all completely new to me :(
• The mission is set up that way, you can leave feedback or press "I haven't learnt this yet" and you will proceed with the next question. It happens to me in many of the cases aswell, so don't feel discouraged because a topic seems unfamiliar. If you want to have fewer situations when unknown topics appear in the missions, you can do more practice in familiar topics. This works great for me when learning a new subject.
• When the series in divided into two parts, would the upper bound of the first part be K?
(1 vote)
• Yes, the partial sum runs from n=1 to k. He is using "upper bound" in a different sense, though, so be careful of the distinction.
• I do not understand why Rk is less than the intergral of f(x) from k to intinity.
(1 vote)
• Rk is the sum of f(n) from n = k +1 to &infin;. He draws the function f(x) on the graph, then at each integer, x= k+1, k+2, k+3, ..., he draws a rectangle with height f(k+1), f(k+2), ..., and width 1, so that their area is equal to their height (since their base is 1). But you can see that every rectangle is under the curve because the curve is decreasing, so that the sum of the areas of the rectangles (which is exactly Rk) must be less than the total area under the curve (which is the integral of f(x) from k to &infin;.).