More Challenging Limits

We have seen in previous pages some fundamental examples that you should know. Here we will discuss some challenging examples. We advise you to first try to find the solution before you read the answer. Good Luck...

Example: For any real number a, define [a] to be the largest integer less than or equal to a. Let x be a real number. Show that the sequence tex2html_wrap_inline168 where

displaymath170

is convergent. Find its limit.

Answer: For any real number a, we have

displaymath174,

or

displaymath176.

Hence, for any integer tex2html_wrap_inline178 , we have

displaymath180.

This implies

displaymath182,

which is the same as

displaymath184.

Since

displaymath186,

we get

displaymath188.

Dividing by tex2html_wrap_inline190 , we get

displaymath192.

Since

displaymath194,

the Pinching Theorem gives

displaymath196.

Example: Let tex2html_wrap_inline168 be a convergent sequence. Show that the new sequence

displaymath200

is convergent. Moreover, we have

displaymath202.

Answer: Set

displaymath204.

Algebraic manipulation give

displaymath206.

Let tex2html_wrap_inline208 . Then, there exists tex2html_wrap_inline210, such that for any tex2html_wrap_inline212 , we have

displaymath214.

Hence, for tex2html_wrap_inline212 , we have

displaymath218,

which implies

displaymath220.

Write

displaymath222.

Since tex2html_wrap_inline224 , then there exists tex2html_wrap_inline226 such that for any tex2html_wrap_inline228 , we have

displaymath230.

Putting these equations together, we get

displaymath232.

So, for tex2html_wrap_inline234 , we get

displaymath236.

This completes the proof of our statement.

Remark: The new sequence generated from tex2html_wrap_inline168 is called the Cesaro Mean of the sequence. Note that for the sequence tex2html_wrap_inline240 the Cesaro Mean converges to 0, while the initial sequence does not converge.

In the next example we consider the Geometric Mean.

Example: Let tex2html_wrap_inline168 be a sequence of positive numbers (that is tex2html_wrap_inline244 for any tex2html_wrap_inline246 ). Define the geometric mean by

displaymath248.

Show that if tex2html_wrap_inline168 is convergent, then tex2html_wrap_inline252 is also convergent and

displaymath254.

Answer: Since tex2html_wrap_inline244 , we may use the logarithmic function to get

displaymath258.

This means that the sequence tex2html_wrap_inline260 is the Cesaro Mean of the sequence tex2html_wrap_inline262. Since tex2html_wrap_inline168 is convergent, we deduce that tex2html_wrap_inline262 is also convergent. Moreover, we have

displaymath268.

Using the previous example we conclude that the sequence tex2html_wrap_inline260 is convergent and

displaymath272,

using the exponential function, we deduce that the sequence tex2html_wrap_inline252 is convergent and

displaymath254.

Example: Let tex2html_wrap_inline168 be a sequence of real numbers such that

displaymath280.

Show that

displaymath282.

Answer: Write tex2html_wrap_inline284 . Then, we have

\begin{displaymath}x_n - x_1 = (x_n - x_{n-1}) + \cdots+ (x_3 - x_2) + (x_2 - x_1) = v_{n-1} + \cdots + v_2 + v_1 \;.\end{displaymath}

In other words, the sequence tex2html_wrap_inline288 is the Cesaro Mean of the sequence tex2html_wrap_inline252 . Since

displaymath292,

the sequence tex2html_wrap_inline252 is also convergent. Moreover, we have

displaymath296.

Remark: A similar result for the ratio goes as follows:

Let tex2html_wrap_inline168 be a sequence of positive numbers (that is, tex2html_wrap_inline244 for any tex2html_wrap_inline246 ). Assume that

displaymath304.

Show that

displaymath306.

Answer:

Below are some more challenging examples. Click on Answer, to get some hints on the solution.

Problem 1: Let tex2html_wrap_inline168 be a sequence of positive numbers (that is, tex2html_wrap_inline244 for any tex2html_wrap_inline246 ). Assume that

displaymath304.

Show that

1.
If |L| <1, then tex2html_wrap_inline318
2.
If |L| > 1, then the sequence tex2html_wrap_inline168 is not convergent.
3.
What can be said about tex2html_wrap_inline168 when |L| = 1?
4.
Use the above to discuss convergence of

displaymath328

where x is any real number.

Answer:

Problem 2: Define the sequence tex2html_wrap_inline168 by

displaymath334.

1.
Show that for any tex2html_wrap_inline246 , we have

displaymath338.

2.
Show that tex2html_wrap_inline168 is decreasing.
3.
Deduce from 1. and 2. that tex2html_wrap_inline168 is convergent and find its limit.

Answer:

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