Proof of Divergence for the Harmonic Series.

In summary, the Harmonic Series is a mathematical series that diverges as more terms are added, and its Proof of Divergence is demonstrated using the Comparison Test. This proof is important because it shows the behavior of infinite series and can also be applied to other series with similar terms. Other methods, such as the Integral Test and the Cauchy Condensation Test, can also be used to prove the divergence of the Harmonic Series.
  • #1
vg19
67
0

Homework Statement


Prove the divergence of the harmonic series by contridiction


Homework Equations


Attached file


The Attempt at a Solution



I understand what they are doing in the first two lines, however, the lines after assuming the series converges with sum S, confuses me. They list the harmonic series and are adding terms in sets of three. I can't see where the next line comes from ( > 1 + 3/3 + 3/6 + 3/9).

Would somebody please be able to help me understand this proof?

Thanks
 

Attachments

  • proof.JPG
    proof.JPG
    26.4 KB · Views: 454
Physics news on Phys.org
  • #2
Remember that 1/2+1/4>2/3?

So, 1/2+1/3+1/4=1/2+1/4+1/3>2/3+1/3=3/3
 
  • #3
Ohhhh...makes sense. Thanks a lot!
 

1. What is the Harmonic Series?

The Harmonic Series is a mathematical series in which the terms are the reciprocals of positive integers. It is written as 1 + 1/2 + 1/3 + 1/4 + ... and is known to diverge, meaning the sum of the series approaches infinity as more terms are added.

2. How is the Proof of Divergence for the Harmonic Series demonstrated?

The Proof of Divergence for the Harmonic Series is demonstrated using the Comparison Test, which states that if two series have the same terms and one series converges, then the other series must also converge. In this case, the Harmonic Series is compared to the series 1 + 1/2 + 1/2 + 1/4 + ..., which is known to converge, thus proving the divergence of the Harmonic Series.

3. Why is the Proof of Divergence for the Harmonic Series important?

The Proof of Divergence for the Harmonic Series is important because it shows that even though the terms of the series decrease, the sum of the series still approaches infinity. This has implications in various fields of mathematics and physics, and helps to understand the behavior of infinite series.

4. Can the Proof of Divergence for the Harmonic Series be applied to other series?

Yes, the Proof of Divergence for the Harmonic Series can be applied to other series with similar terms, such as the series 1 + 1/2 + 1/3^2 + 1/4^2 + ... This series also diverges due to the Comparison Test, as it can be compared to the Harmonic Series.

5. Are there any other proofs for the divergence of the Harmonic Series?

Yes, there are other methods for proving the divergence of the Harmonic Series, such as the Integral Test and the Cauchy Condensation Test. These methods use different techniques, but ultimately arrive at the same conclusion of the series diverging.

Similar threads

  • Calculus and Beyond Homework Help
Replies
29
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
658
  • Calculus and Beyond Homework Help
Replies
2
Views
746
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
85
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
3K
Replies
3
Views
898
  • Calculus and Beyond Homework Help
Replies
22
Views
2K
Back
Top