Exercises: Series – Serlo

Telescoping series Bearbeiten

Exercise

Investigate, whether the following series converge. If so, compute their limits.

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  

Hint regarding subtask 3: We have  . Why?

Hint regarding subtask 5: We have  .

Solution

Subtask 1: This is a telescoping series with  . Taking a look at the partial sums we get

 

As   diverges, the series diverges as well.

Alternative solution: One can easily find a lower bound for the partial sum sequence:

 

As   (harmonic series),   is not bounded from above/below. Hence, the series diverges.

Subtask 2: We have

 


Obviously, this is a telescoping series with  . We get:

 

Subtask 3: Take a look at the hint. We get

 

This is a more generalized version of a telescoping sum. The first and last two summands do not cancel:

 

Subtask 4: We have

 

We get the following telescoping series:

 

Subtask 5: Take a look at the hint! We get

 

Hence, we can calculate the series using two telescoping series:

 

Alternative solution: It holds that

 

Using the properties of telescoping series, we get:

 

Solution 6: It holds that

 

It follows that

 

Geometric series Bearbeiten

Exercise (Geometric series: Convergence and limits)

Investigate, whether the following series converge. If so, compute their limits.

  1.  
  2.  
  3.  
  4.  
  5.   with   for   being even and   for   being odd.
  6.  

Solution (Geometric series: Convergence and limits)

Subtask 1: We have

 

Subtask 2: As  , this series diverges.

Subtask 3: The series   converges. Using the computation rules for series, we get

 

Subtask 4: The series   and   converge. Using the computation rules for series, we get

 

Subtask 5: The series   and   converge. Using the computation rule for series, we get

 

Subtask 6: The series   and   converge. Using the computation rules for series, we get

 

Harmonic series Bearbeiten

Exercise (Harmonic series)

You may assume that   converges and that   holds.

  1. Explain why the series  ,   and   converge.
  2. Compute   and  .

Solution (Harmonic series)

Subtask 1:

1st series: The partial sum sequence   is monotonously increasing as all summands are positive. Futhermore,   is bounded from above as

 

Hence   converges.

2nd series: We know that   converges. Using the limit theorems for series, we get  . Hence, this series converges.

3rd series: As the series   converges absolutely, it converges.


Subtask 2:

1st series: We have

 

It follows that

 

2nd series: We have

 

Remark Bearbeiten

Analogously, for the generalizes harmonic series   with   we can show:

  •  
  •  


Exercise (Alternating harmonic series)

You may assume that   converges and that   holds.

Explain why the series   converges and compute its limit.

Solution (Alternating harmonic series)

  • Convergence: We will show that the series converges absolutely. In the article about absolute convergence we have proven that this implies convergence. Let  . As all summands are larger that zero,   increases monotonically. Furthermore, we have
 .

Hence,   is bounded and converges.

  • Limit: We have
 

e-series Bearbeiten

Exercise (e-series)

Explain why the following series converge and compute their limits:

  1.  
  2.  

Solution (e-series)

Subtask 1: The partial sum sequence   increases monotonously and is bounded from above as

 

Hence, the sequence   converges.

Furthermore, we have

 

Alternative solution: Via telescoping sum. We have

 

Subtask 2: The partial sum sequence   increases monotonously and is bounded from above as

 

Hence, the sequence   converges.

Furthermore, we have

 

Rearrangement theorem for series Bearbeiten

Exercise (Rearrangement of alternating harmonic series)

The alternating harmonic series

 

and

 

converge to   resp.  . Show that the following rearrangements converge to the limits given.

  1.  
  2.  
  3.  

Hint regarding subtask 2: Start off showing that   with   being the  -th partial sum of the alternating harmonic series and   being the  -th partial sum of the rearranges series.

Solution (Rearrangement of alternating harmonic series)

Subtask 1: With   and   being the partial sums of the alternating harmonic series and the first rearrangement, we have:

 

  converges and hence,  tends to  . Thus,   converges and   tends to   as well.

Subtask 2 We have

 

As   and   tend to  ,   tends to  . We can conclude that   converges and   tends to  .

Subtask 3: As  , the series   converges absolutely. Using the rearrangement theorem for absolutely convergent series, we get that every rearrangement of the series converges and tends to the same limit. Hence, the given rearrangement tends to  .

Exercise (Rearrangement of convergent but not absolutely convergent series)

Prove the following statements: Let   be a convergent -but not absolutely convergent - series. Then there exists a rearrangement of the series that...

  1. diverges but not to   or  .
  2. converges to an arbitrary  .