Stumped on \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}

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In summary, the conversation discusses whether a specific type of series, represented by {a0, a1,...}, will diverge or not. The initial thought was that it would not diverge, but after trying different tests, the possibility of it diverging was considered. It is then mentioned that a specific example, where a0=1 and an=n for n>0, would make both series diverge. The question is then clarified to ask if all series of this form will diverge, and it is stated that previous examples only cover a certain type of divergent series. The conversation then mentions that requiring aan to make sense would result in an increasing, unbounded sequence of positive integers, which would lead to divergence. However, a
  • #1
Nexus[Free-DC]
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This thing has me tearing my hair out:

Let {a0, a1,...} be a sequence such that
[tex]\sum_{n=0}^{\infty}{\frac{1}{a_{n}}}[/tex] diverges.

Does [tex]\sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}[/tex] diverge?

My first instinct was to say no, but then I couldn't find any counterexamples. Now I am thinking it might actually be true but it has defied all the tests I've tried. Any ideas?
 
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  • #2
Let a0=1 and an=n for n>0. Both series are the same and diverge.
 
  • #3
Sorry, I guess I wasn't clear enough. Do ALL such series diverge? I already know all series of the form an=kn+c do since aan = k(kn+c)+c=k^2n+kc+c, but that doesn't cover all divergent series.
 
  • #4
Doesn't your requirement that aan make sense require that an be an increasing, unbounded, sequence of positive integers- and so any subsequence will diverge?
 
  • #5
If an=n2, both series converge. It looks like it would be hard to construct an example where the first diverges and second converges.
 

1. What is the meaning of the equation \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}?

The equation \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}} represents an infinite series, where the value of each term is calculated by taking the reciprocal of the value of a_{a_{n}}. The equation is often used in mathematics and physics to describe phenomena that involve infinite sums of terms.

2. How do you solve the equation \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}?

The solution to the equation \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}} depends on the specific values of a_{a_{n}}. In some cases, the infinite series may converge to a finite value, while in other cases it may diverge to infinity. To solve the equation, one must first determine the behavior of the series and then apply appropriate mathematical techniques to calculate its value.

3. What is the significance of the value of \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}?

The value of \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}} can have various meanings depending on the context in which it is used. In mathematics, it can represent the limit of a sequence or the value of an integral. In physics, it can describe physical quantities such as energy or probability. The significance of the value also depends on the specific values of a_{a_{n}} and the problem being studied.

4. Can the equation \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}} be applied to real-world problems?

Yes, the equation \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}} can be applied to real-world problems in mathematics, physics, and other fields. It is commonly used to describe and solve problems involving infinite series, such as calculating the value of a continuous function or determining the behavior of a physical system over time.

5. Are there any limitations to using the equation \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}?

As with any mathematical equation, there are limitations to using \sum_{n=0}^{\infty}{\frac{1}{a_{a_{n}}}}. One limitation is that it can only be applied to problems that involve infinite series. Additionally, the equation may not provide an exact solution in all cases, as some infinite series may have no closed form solution. It is also important to consider the convergence or divergence of the series when using this equation to solve problems.

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