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More Challenging Limits

 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.

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