Find the sum to infinite series

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Homework Help Overview

The discussion revolves around the infinite series involving the inverse cotangent function, specifically the terms cot^-1(3), cot^-1(7), cot^-1(13), and so forth. Participants are attempting to analyze the series and its convergence properties.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • One participant attempts to express the nth term of the series as t_n = cot^{-1}(2^n + (2n-1)) and is exploring the limit as n approaches infinity. Another participant questions the generalization of the sequence 3, 7, 13, and whether it is correctly represented as 2n + (2n-1).

Discussion Status

The discussion is ongoing, with participants providing feedback on the presentation of the terms and questioning the assumptions made about the series' structure. There is no explicit consensus yet, but participants are engaging with each other's ideas and clarifying points of confusion.

Contextual Notes

There are indications of confusion regarding the notation used for the inverse cotangent function, as well as uncertainty about the general form of the series terms. The original poster expresses difficulty in calculating limits, which may affect the progression of the discussion.

utkarshakash
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Homework Statement


cot^-1 3 + cot^-1 7 + cot^-1 13+...


Homework Equations



The Attempt at a Solution



I first tried to write the nth term of the series

t_n = cot^{-1}\left( 2^n + (2n-1) \right)

Then I tried to calculate the limit as n→∞. But I simply can't do that. I mean I don't know how to calculate the limit.
 
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utkarshakash said:

Homework Statement


cot^-1 3 + cot^-1 7 + cot^-1 13+...


Homework Equations



The Attempt at a Solution



I first tried to write the nth term of the series

t_n = cot^{-1}\left( 2^n + (2n-1) \right)

Then I tried to calculate the limit as n→∞. But I simply can't do that. I mean I don't know how to calculate the limit.

Your presentation is very confusing; use parentheses, like this:
cot^(-1)(3) + cot^(-1)(7) + ... or use "arccot" instead of "cot^-1".
 
Ray Vickson said:
Your presentation is very confusing; use parentheses, like this:
cot^(-1)(3) + cot^(-1)(7) + ... or use "arccot" instead of "cot^-1".

OK Here's it

cot^{-1} 3 + cot^{-1} 7 + cot^{-1} 13+...
 
utkarshakash said:
t_n = cot^{-1}\left( 2^n + (2n-1) \right)
It's not at all obvious to me how 3, 7, 13.. is supposed to be generalised. Do you know that it is intended to be 2n+2n-1, or are you guessing?
 

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