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I was solving some questions of analysis, and after long time, I came across analysis, I didn't get any direction.

Q. Discuss the convergence of following series.

(1) $\sum_{n=1}^{\infty } [(n^3+1)^{1/3}-n]$.

(2) $\sum_{n=1}^{\infty } [(n^4+n^3+n^2+n+1)^{1/4}-n]$.

Any hint for checking convergence?

What I immediately did was following: $f_n=(n^3+1)^{1/3}-n \leq (n^3+n^3)^{1/3}-n=(2^{1/3}-1)n=g_n$ but series $\sum g_n$ do not converge, we can't say about given series. I don't get any other direction.

Beginner
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  • Although I ask for hint (but not complete answer), I don't understand what is the reason for downvote? I can see many very elementary group theory questions but I don't think to dovnvote/upvote immediately, unless I don't know background of person who asks it. – Beginner Nov 03 '17 at 11:36
  • Do you know anything about Taylor? or should we try to do it in another way? – J. Sadek Nov 03 '17 at 11:40
  • Sorry, I have not remembered it well; but I will see it and try to apply. Let me try. – Beginner Nov 03 '17 at 11:41
  • "I don't understand what is the reason for downvote" There are two immediate things about your question that I know attracts downvotes: 1) You haven't really shown any of your own work or thoughts; you have just written down the problem and ask for help. If you had had at least a line or three of calculations you have tried, that would placate most of them. 2) You are asking two questions in one. This makes it difficult to write an answer. Say, for instance, that I know how to solve the first but not the second. Would that be fitting to post as an answer? It's not so easy to tell. – Arthur Nov 03 '17 at 11:48
  • @Arthur: I agrree on (1). I have edited question. (Also I am coming to this after long gap since I usually work in algebra; I don't find immetiately direction) – Beginner Nov 03 '17 at 11:53

1 Answers1

1

$(n^3+1)^{1/3}-n=n\left(\left(1+\frac 1 {n^3}\right)^{\frac 1 3}-1\right)=n\left(\frac 1 {n^3} + o\left(\frac 1 {n^3}\right)\right)=\frac 1 {n^2} + o\left(\frac 1 {n^2}\right)$ which are both general terms of convergents series.

Hence 1 is convergent.

I'll let you do 2

J. Sadek
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