How to show a sequence converges

How to show a sequence converges

What we need to do:

First we must decide what limit we think the sequence is convering to. Without that, we can't possibly make the right estimates. If we're told what the sequence converges to, well and good. If we're not, we absolutely have to reach the answer by some sort of creative guesswork. Just writing down an "a" that we can't specify and using it as the limit, won't cut it.

Then, we need to take an arbitrary positive d make estimates on the quantity x_n-a with the aim of finding an N such that |x_n-a| is less than d for all n above N.

What we say:

Let d> 0 be given...

...and we find an N. Thus, |x_n-x|< d for all n> N.

What we want to do:

Use the information from the particular problem to estimate |x_n-a|. Find a point beyond which this is alwys smaller than d.

Example:

Question

Solution

(To see the solution to the second part, click here

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Analysis WebNotes by John Lindsay Orr.
Comments to the author: jorr@math.unl.edu

Last modified: June 13, 1995