Is my proof of this sequence's divergence good enough?

In summary, the conversation discusses the divergence of the infinite series Ʃ n=1 to infinity of cos(n∏). Through the use of the nth term test and rewriting the term as (-1)^n=an, it is shown that the series does not converge as the limit of the general element's sequence does not approach zero. However, the language used may be considered informal and could benefit from a more precise explanation.
  • #1
mathnoobie
63
0
Ʃ n=1 to infinity of cos(n∏)
letting an=cos(n∏), I rewrote this as (-1)^n=an.

Using the nth term test i let the limit as n->∞ go to infinity. This value bounces back and forth between positive and negative, but I know clearly the value =/= 0, therefore it diverges by the nth term test.

Is there anything I should add to my proof?
 
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  • #2
mathnoobie said:
Ʃ n=1 to infinity of cos(n∏)
letting an=cos(n∏), I rewrote this as (-1)^n=an.

Using the nth term test i let the limit as n->∞ go to infinity. This value bounces back and forth between positive and negative, but I know clearly the value =/= 0, therefore it diverges by the nth term test.

Is there anything I should add to my proof?



Looks fine to me, although with a somewhat folkloric language as "bouncing back and forth" and "n-th term test", which I don't what

is. Perhaps you meant simply that the series doesn't converge as its general element's sequence doesn't converge to zero, which is a necessary condition.

DonAntonio
 

1. How can I determine if my proof of a sequence's divergence is sufficient?

To determine if your proof is sufficient, you should first make sure that you have clearly stated and proven your assumptions, definitions, and theorems used in your proof. Next, check if your proof follows a logical and well-structured argument. Finally, compare your proof to established and accepted proofs of similar problems to see if they are similar in approach and rigor.

2. What are the common mistakes to avoid when proving a sequence's divergence?

Some common mistakes to avoid when proving a sequence's divergence include making assumptions without justification, skipping steps in your proof, and using incorrect or unclear notation. It is also important to avoid circular reasoning and to thoroughly check all calculations and algebraic manipulations.

3. Is it necessary to use mathematical symbols and notation in my proof?

While mathematical symbols and notation can make your proof more concise and clear, it is not always necessary. As long as your proof is well-written and logically sound, it can be presented in plain English. However, using mathematical symbols and notation can improve the readability and rigor of your proof.

4. What should I do if I am unsure about a step in my proof of a sequence's divergence?

If you are unsure about a step in your proof, it is important to carefully review your assumptions, definitions, and theorems to see if they support your argument. You can also consult with other mathematicians or experts in your field for feedback and suggestions. It is better to address any uncertainties or potential errors early on in your proof rather than having to backtrack later.

5. Can I use examples or counterexamples to support my proof of a sequence's divergence?

Yes, using examples or counterexamples can be helpful in illustrating and supporting your proof. However, it is important to note that examples and counterexamples should not be used as a substitute for a rigorous proof. They should be used to supplement and enhance your proof, not replace it.

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