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Dick said:For the first one, you are given the first series converges. That means a_n->infinity. Think comparison test. 1/(a_n-x)<1/(a_n-y) if x<y<a_n. How about choosing y=a_n/2?? Can you justify that? For the second one, write it as [(n+1)^n/n^n]*[1/n^z]. The first factor has a limit. What is it?
asi123 said:This are my thoughts, is this right?
Dick said:Why do you think 1/(a_n-2) converges? Shouldn't you state a reason?
asi123 said:If 1/(a_n-2) converges, than why shouldn't 1/(a_n-2) converge? I mean if a_n -> infinity than I don't think that 2 will bother him, no?
Dick said:No, the 2 won't bother him. But you still have to show that. Set up a comparison test with something you know converges. Review my hint about this one.
Convergence refers to the behavior of a sequence or series as its terms approach a specific value or limit. In other words, it is the tendency of a sequence or series to get closer and closer to a certain value as more terms are added.
Convergence and divergence are opposite behaviors of a sequence or series. Convergence occurs when the terms of a sequence or series approach a specific value or limit, while divergence occurs when the terms do not approach a specific value or limit and instead increase or decrease without bound.
To prove convergence, you can use various methods such as the squeeze theorem, the monotone convergence theorem, or the ratio test. These methods involve analyzing the behavior of the sequence or series in question and determining if it approaches a specific value or limit.
The radius of convergence is a measure of how quickly a power series approaches a specific value or limit. It is the distance from the center of the series to the point where the series converges. The larger the radius of convergence, the faster the series converges to its limit.
The radius of convergence can be found by using the ratio test or the root test, which determine the values of x for which the series will converge. Additionally, the radius of convergence can also be found by using the Taylor series expansion of a function, which gives the interval of convergence for that specific function.