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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  where

is convergent. Find its limit.

Answer: For any real number a, we have

,

or

.

Hence, for any integer  , we have

.

This implies

,

which is the same as

.

Since

,

we get

.

Dividing by  , we get

.

Since

,

the Pinching Theorem gives

.

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

is convergent. Moreover, we have

.

.

Algebraic manipulation give

.

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

.

Hence, for  , we have

,

which implies

.

Write

.

Since  , then there exists  such that for any  , we have

.

Putting these equations together, we get

.

So, for  , we get

.

This completes the proof of our statement.

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

In the next example we consider the Geometric Mean.

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

.

Show that if  is convergent, then  is also convergent and

.

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

.

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

.

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

,

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

.

Example: Let  be a sequence of real numbers such that

.

Show that

.

Answer: Write  . Then, we have

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

,

the sequence  is also convergent. Moreover, we have

.

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

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

.

Show that

.