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
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
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.
Exercise (e-series)
Explain why the following series converge and compute their limits:
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