Find the sum of each series. ∑n=1^∞(tan^-1(n)-tan^-1(n+1)) |...

Name: Problem 52, Chapter 9: University Calculus: Early Transcendentals
Uploaded: 2020-05-02T16:02:10-08:00
Duration: 5 min 12 s
Channel: Kevin Shryock
Description: Find the sum of each series. ∑n=1^∞(tan^-1(n)-tan^-1(n+1))

Key Concepts

Telescoping Series

A telescoping series is one in which consecutive terms cancel each other out, leaving only the first and the last terms when the series is summed. This characteristic makes it easier to calculate the overall sum by focusing on the few terms that do not cancel, typically evaluated in the limit as n approaches infinity.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the arctan (or tan?¹) function, are used to find the angle whose trigonometric function equals a given value. Their continuity and limiting behavior often provide insight into summing series involving these functions, especially when combined in a telescoping structure.

Series Summation Techniques

Series summation techniques include methods for summing infinite series, such as analyzing term cancellation, computing partial sums, and taking limits to determine convergence. These techniques are vital to ensure the proper evaluation of series, particularly when dealing with infinite or telescoping sums.

Transcript

00:01 Hello everybody and welcome back to numerant.

00:02 This is kevin chirac.

00:03 Let's consider this infinite series that is denoted by the series summation from n equals 1 to infinity of inverse tangent of n plus 1.

00:16 And let's begin to expand this out and see a little bit of what we get when we play.

00:20 So if we put in an n value of 1, that's going to be the inverse tangent of 1 for our first part of our first term.

00:30 And then our second part of our first term is going to be the inverse tangent.

00:34 Of two.

00:36 Simple enough.

00:37 Let's now add that to the next term, which is going to be inverse tangent, or some people call it arc tangent, arc tangent of two minus arc tangent of three.

00:54 And we're just going to continue to do this forever, never, never, never.

00:59 And we can even get to an emth term, which would just be the arc tangent of m minus the arc tangent of m plus one.

01:13 And then since this is an infinite series we would continue adding up from there.

01:19 But if we were to consider what the partial summation is up to that m value, that's almost gonna be as if we cut it off, or it's gonna be exactly as if we cut it off right here and we kind of ignore all of those additional dot dot dots that are gonna occur.

01:32 So let's take a look at what happens just up to that m value.

01:35 Well, up to that m value, i have a bunch of cancellations that can occur because i've got an arc tangent of two and a negative arc tangent of two.

01:41 So those can cancel away.

01:43 And then this is like, going to cancel away or in fact you can deduce it will cancel away with the parenthetical that's immediately after that you can have a bunch of cancellations that occur in here and you can even see that if we're to add in maybe our term immediately preceding that nth term so it would be the arc tangent of m minus one minus the arc tangent of m minus one plus one so that just be m and there we are we see that this m value is going to cancel away and this m minus one value would have canceled away with the one preceding it...

Please give Ace some feedback